Great for supporting land forces, terrible for pure naval combat

Early modern rocket artillery weapons as seen in WW2 were area saturation weapons for supporting ground troops at the front. The key here is that the rockets were all unguided. On average, each individual rocket had abysmal accuracy, so they had to be fired in large quantities to compensate. The resulting bombardments are just fine for destroying soft skinned vehicles, messing up static defenses and the infantry usually manning them. However, they are pretty crap against proper tanks or combat ships; a near miss that would injure and incapacitate a foot soldier does nothing(or near enough) to such armoured vehicles. Also, remember that the concept of "suppression" in infantry combat doesn't really apply to tanks and ships.

For pure ship-on-ship combat, IMO what this concept amounts to is essentially a later and worse carronade in an era where such weapons have less relevance. While massed rocket salvos may do potentially greater damage than regular naval guns, to make use of them you have to close to what is effectively knife fighting distance in the naval context while the enemy basically gets to shoot at you uncontested. Trying to fight another ship at standoff distance with unguided rockets results in a mostly harmless fireworks display

Then there's also logistics. It's fairly obvious that large salvos of rockets fired rapidly means such weapon systems consume ammo at a much faster rate than guns do. For ships who have to operate at sea for long periods without resupply, this is a significant problem. Running dry after just one or two engagements is a potentially fatal flaw. This is less of an issue for land-based artillery units who have readier access to resupply.

The reality check verdict here is basically: dubious. Might be considered for use as a secondary weapon on larger vessels should said navy wish to diversify their ships' capabilities, but will never fully replace regular naval guns without the invention of guidance systems for the rockets.

Possible uses: There are still ways to employ such weapons on ships, albeit limited. The first way I can think of is a bolt on "one shot wonder" upgrade like the Calliope or the Stuka zu fuss. The single massive barrage these would provide would most likely be used in the opening phase of a major offensive like the Gallipoli Campaign, kicking off set piece battle(s).

The second would be launcher batteries hidden in the hull of a ship instead of tacked on and visible externally. This method would be a "gotcha" used as an ace in the hole should fighting close to spitball distance. Rockets used this way would be fired straight at the enemy rather than in a parabolic arc when used as artillery.

Pro

• Range. Both the US Army and Navy have extended range rocket assisted projectiles (RAP). The Army 155mm howitzer normally has a range of 22 km, but up to 30 km with the RAP. The RAP also can make some favorable shots at high elevation (i.e. the gun is pointed nearly straight up), which was a particular issue shooting over mountains in Afghanistan. The Navy's LRLAP is a much larger rocket, with a range of 150 km or more. Otherwise, the Navy's 5" gun can manage around 25 km range.

Con

• Accuracy. Despite rockets being around for a long time (like, since the Medieval Chinese), they weren't used in Naval guns until relatively modern times. That is because extended range RAPs are very inaccurate. Launching rocket shells 25 miles away is tantamount to throwing valuable ordinance into the ocean. When they started putting rockets on naval shells, they called them guided missiles....because they were guided. The modern RAP has a GPS guidance system and flips some fin stabilizers out to ensure it gets where it is going.




• Cost. Can you believe that the Navy's new LRLAP (linked above) costs a million dollars per shot? I can't either. John McCain is dead, no one is holding the Pentagon responsible any more. Please write to your congressman.

COST

Its much cheaper to put a projectile in a tube and blast it out with explosives.

Rockets require more effort in assembly to craft the shape and motor. This can also result in a waste in size as well which further reduces cost benefits.

Now missiles on the other hand have the ability to alter their course which opens up whole new tactical capabilities.

Rockets have a lower explosives payload than gun/bomb shells because they need to include the propulsion which is ordinarily provided by the gun. The propulsion unit is probably not reusable, making rockets expensive. Using guns should be much more economic per delivered amount of explosive or just momentum.

Using rockets potentially gives you a higher density of a single wave. That is, if you can mount all the rockets you have somehow, you can probably fire them all at once, while with a gun you are probably much more limited with your fire rate.

Rockets at this time were not guided. However rockets because of their on-board maneuvering capabilities can in principle be guided (german V flying bomb), while bomb shells are purely ballistic.

What you would need would be conditions where the advantage of rockets (high densities of waves) is increased and the disadvantage (comparably low amount of delivered explosives) is decreased.

Decreasing the disadvantage could be achieved by making explosives super efficient, that is already a small amount of explosive is enough to kill the ship of the enemy.

However, wouldn't that make gun shells also much more deadly? Yes, it would. But we still have the low/high density thing.

How can a high density of a wave help? Counter measures like trying to shot down incoming shells/rockets were not invented really and would probably not work anyway at the incoming speeds and the existing technology.

A high density can help if the aiming accuracy is generally low. The aiming accuracy can be made lower if the battle distance is large. Make it so large that only a dense wave of super efficient rockets has a significant chance for a hit while every hit has a very high chance of killing the target.

Using guns under this circumstances is like using a sledgehammer to crack a nut. Very exhausting and not very economic and just less effective than rockets.

Maybe you could even invent a primitive guiding system for the rockets. Some kind of analogue electronic circuit logic. Simple but a bit better than purely ballistic rockets (what about strong chaotic, turbulent winds which need to be compensated).

But why should the distance be large? Well, the super efficient explosives will also help making the range longer if you use them as propellants.

Summary: Use super efficient explosives. With super efficient explosives the one who shoots first in the right direction wins.

Example scenario:

Two hostile fleets of rocket carriers are 200 nautical miles apart. It's very windy and rainy. They have small scout ships between them that should locate and communicate (that is actually the key here) the position of the hostile fleet as quickly as possible. The one fleet that can fire first all the rockets they have toward roughly the right position will win the battle (unless they are very unlucky). However, killing the enemy is really important otherwise he might come into range of our homeland cities and send his rockets there (and the cities locations are known, so they are an easy target).

Thea main reason why the Soviet's Katyusha is successful is they fired hundreds of these towards the targets position, which negates the rocket's issue regarding accuracy. But Reloading time is a pain.

Now in a naval battle, if you going to use rocket use thousands of them, and pray that they dealt critical damage to your enemy, if not, then pray that they treat you well once you surrendered.

Rocket vs gun is an interesting question. Apart from the very well known example of land warfare, rockets were also used in the air with one major advantage. A fighter with 24 rockets of a good caliber was lighter than the same plane with a canon able to fire the same caliber. Also, while being less accurate, these german planes were able to launch a salvo and cripple the thin armor of a bomber, and cause considerable dammages in a single row.

For me it is a use case for a small attacker, to deal dammage to a bigger yet not very well armored opponent quickly and then move. That's a weapon of choice for lights ships at close quarter range : this way they can cripple a big one on its weak point, or destroy another ship of its size. However, bigger hulls would be pretty useless when fitted with rockets. They would be easier to target, and doesn't really need the big size required to sustain bigger guns. Thus battleships would need a total design overhaul, they would probably be lighter, and used as combat platforms with the few guns your universe allow.

For me naval warfare would then be more oriented on close range combat, thus using probably more small ships (corvettes, frigates, destroyer) than medium and heavy ones (cruisers, battlecruisers or battleships). If there is no rubber I assume planes aren't an option in your world so I think there are no aircraft carriers (or no modern aircrafts). I imagine fleets would be made of hordes of small ships able to close in easily while dodging any volley of small guns the enemy would be foolish enough to fire, to be in a good position to use their close quarter batteries. All of these small ships might be supported by fast battleships, or battlecruisers, ships with slightly better armor than small ones (enough to sustain fire from rockets), loaded with short range AP guns.

I think AP rockets aren't present in your universe, as they weren't really existing in the 20's. So medium/heavy ships could carry the armor piercing shells, the ones that could engage fights at mid range, while the small ships would mostly use HE rockets to wreck havoc on other light ships / try to mess with the big guns.

Indeed rockets can deal great dammages to the deck of a ship ! As we knew ship in the 20, the crew on the deck could suffer loses while maning small caliber gun (i.e : with no armor). Thus I imagine all of your ship to be much more armored on the deck. A lot of ships deck were wooden made, I imagine with the use of deadly HE rockets that could be sent ten by ten, you'd need to protect against it, have better fire control team, hardened decks, and most likely side armor for your crew. Otherwise your sailors would die like fly in short range fights.

Last but not least, with less deadly naval guns, closing the distance would be that much easier. Which means torpedoes used as a main weapon becomes a bigger threat. With no air cover, most weapons being inaccurate, I imagine easily specialized torpedoe ships closing the distance to launch deadly torpedoes salvos. These were quite accurante, and could have a payload even bigger than naval guns shell. Also the first remotely controlled torpedoes were proof tested after the first world war, and worked (yet they were not massively deployed).

To protect against these torpedoe ships, that could be very vulnerable to HE rockets, escort ships could then carry devices similar to Hedgehog launchers : a bunch of 12/24/36 rockets to launch in a big cluster all at once, to cover a wide area in a "spray and pray" tactic, with deadly effect for small ships.

To sum up :

In terms of strategies this would mean :

• The use of swarm of ships instead of bigger and bigger ones (so no
  Yamato)


• An increased need for good escort techniques and screening,
  as short ranged weapons are deadlier than ever.


• A different concept of task forces, as capital vessels would be closer to screening ship


• An increased efficiency for submarines (no plane to spot them or to
  use depth charges, less deadly surface weapons against their all
  metal armor)


• A biggest difficulty to prepare for any naval invasion.
  Think like Overlord, with water mines, torpedoes, and land artillery
  firing... but ships that can't retaliate from a long range, with no
  air cover to help. Well done Germany, you won this round.

In terms of design :

• More armor on the side, and the deck, less armor needed for turrets or other critical parts of a ship.




• Lighter and faster capital ship, think more like heavy cruisers (Algerie), battle cruisers (Hood, Dunkerque), than super battleship (Yamato)



• Ships with absolutely no AA, so more space to fit more weapons ! Weapons that are by the way less heavy thus making it easier to have a quicker and better armored ship.


• Torpedoes for everyone ! Battleships had torpedoes, but didn't make extensiv use of these. In this setting, they could use it.


• More interest to use torpedoe bulge for ships.


• Faster development of bigger torpedoes, and perhaps sea mines too, as these are now the biggest threat for a ship

There are a lot of other changes, in technology that I can't cover here ! But I think sonar, and radar would be that much important, as much as developping new propellants (so better developpment in chemistry and industry in general) while alloy production would be that less usefull. There are a lot of things on which you could write and imagine.

The other answers here assume they're taking rockets designed for launched from land, to land; and planting them on ships. In this case, the rockets are launched up to avoid obstacles, and then have to drop their altitude to hit their target.

This clearly has an issue with accuracy, but if you know you're going to be targeting things on the sea, then you'll use a different projectile path. That's just the way engineering works - if you can make a small edit to something existing to make it better in a different environment - you make that edit.

Looking at a modern anti-ship missile, the exorcist missile is one that cruises at 50ft or so... which means it can take a very shallow decline towards the water, allowing it to hit a much larger section of the ship than if it was approaching from 500ft. (It's also guided, which helps).

With a redesign of the rockets, you may be able to have them skim across the water; essentially like a speedboat. This would make them comparable to torpedoes (or speedboats with bombs on - still used largely by terrorists today); and something that in the 1920's would really be something you wouldn't want to see heading towards your ship.