A few battleship possibilities
I can think of several ways to keep battleships around and in service into the early 21st century (or bring them back) in different roles.
1. If one or more major powers considers it likely that an invasion of a hotile beach may be needed, or that sites near the coast may need to be dealt with severely, they could be kept around for naval gunfore support. The USN still has 2 in reserve (Iowa and Wisconsin) but I'd be surprised to see them take to the seas again. The navy also has the right to reclaim 6 others that are museum ships, althought one is approaching the century mark.
In addition, a battleship can be downright intimidating in a way that a carrier can't. One look says, "Danger, I'm the baddest thing on the block."
Also, if bombardments are needed often, a battleship's shells are much cheaper than missiles, and its opperations don't result in lost aircraft. A casualty-sensitive military, where 2 or 3 planes lost is a cause for public outcry, would find this handy.
Also, remember--an airstrike drops its load and goes away. You've got some time before they can come back. If some enemy troops survive, they can start shooting again. With a battleship off shore, any enemys that call attention to themselves are only about a minute away from a 16" shell or 9.
2. Had there been no major wars after World War One, the tempo of military technology may have slowed. Without proof in battle, the battleship would still be around, IMHO, but with missiles as well as guns, and deck armor that would be scarey. The military mind is slow to discard a weapons system until it's proven that it is, in fact, useless.
3. By the 1960's, one reason that there was no need for battleships was simple--no one besides the USN had them. A battleship's purpose is to neutralize an enemy's battleships and secure the sea. With no enemy battleships, carriers can do this as well (and no one had a carrier force that could threaten the USN. I suspect that if a potential enemy had battleships, the USN would retain them as well. Those ships are hard to sink, and a carrier would have to secure control of the sky to do it. Carriers were often sunk in the face of enemy combat air patrols, but the battleships that were sunk by air power did not have effective air cover. I'd expect the fast battleship/carrier combined battlegroup to be in business. The battleship provides AA defence at first, then the battleship goes in harm's way after the enemy's air is reduced.
4. I worked on a "Return of the Battleship" story a while ago, but my focus turned elsewhen, ands it sits on my hard drive for now. I expect to come back to it eventually.
The basic premise is simple. There is a breakthrough in laser technology that leads to a few imeadiate results. Lasers become militarily useful at modest power ranges. They can punch a hole in a thin skinned missile or aircraft, and are deadly accurate. More power is unlikely in the near future, but may happen.
As a result, missiles can't attack well defended targets--fleets with a laser AA cruiser or two, well protected land targets, etc.
A large shell, however, has much solider construction. A laser hit simple won't be enough to damage it. The big gun is back...
And with the return of the big gun comes the return of heavy armor. Sounds like a battleship to me.
Incidently, the new lasers lead to workable fusion power.
5. Another possibility is more limited, and quite unlikely, but concievable to me. Suppose the South American nations resumed their rivalry after WWII? They almost start an arms race, but can't afford it, and agree to no new construction for a period of time. Their old battleships get modernized again and again. When the treaty lapses, none of them want to start a new arms race, and so keep the old ships up. (Perhaps Chile is allowed to buy one of the 12" Italian battleships to even things up, or perhaps simple gets a surplus American cruiser to compensate for the Ciliean ship's bigger guns.)
This continues to the present day.
Incidently, Margeret Thatcher would have to be very careful if Argentina had two fully modernized battleships available for the Falklands...
There's a few spur of the moment thoughts.