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
- 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
- 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.