Peacetime Navies are normally conservative institutions that are adverse to developing revolutionary new weapons systems. If the WW2 Pacific conflict had never happened, it's likely that battleships would have been around for a lot longer and large carriers would have taken more time to appear.
Even after WW2 though, the case that battleships were obsolete is overstated. The main reason the United States Navy didn't build battleships after WW2 is because there were no enemy battleship fleets left in the world to counter. The battleships themselves were still a threat and remained so at least until aircraft could be relied upon to take off and fight at night and in any weather (this probably happens in the late 1950's).
For example, if the Soviets has chosen to build a battleship fleet during the early stages of the Cold War (they had plans to do this but didn't follow through) then it's very likely the United States and Great Britain would have built new battleship fleets to match them.
It's also worth noting that even today most Navies don't build large carriers. The United States Navy, with its mission of "Global Power Projection", is a special case. For most Navies carriers are far too expensive and as they only plan to fight close to home they can rely on land based aircraft. 527
So perhaps, if there is no there is no global super power in your story, but only regional powers, then there might be fewer carriers.
In some ways battleships are still with us as ship size classifications have undergone significant inflation over time. A modern day destroyer for example is of a similar size to an early WW1 battleship and some big cruisers are as big as a WW2 battleship (although without armour they are only about half as heavy) .
If you specifically want to have big ships with heavy amour and large caliber guns then it's difficult to see them still providing the main fighting power of a fleet into the 21st Century. Guns lack the range and guidance capabilities of missiles, and armour is practically useless against torpedoes and mines (when an explosion occurs underneath a ship, the weight of the vessel acts against itself to break the ship in two).
However it's not inconceivable that in you world submarines, torpedoes and mines have been banned or heavily restricted by arms limitation treaties (some people actually did want to ban submarines then they first appeared as they were seen as being an "underhand" way of fighting). There have certainly been naval arms limitation treaties in the past, and even during the Cold War, there were certain types of weapons that both sides refrained from building, even though they would have been very useful - e.g. spaced based missile systems. Perhaps with this treaty in force you'd have big capital ships with enormous missile batteries that could also conceivably have armour as well as active anti missile defenses. Maybe later on in the 21st Century they would have electromagnetic rail guns. 527