In a post WW2 environment, missiles are to battleships what planes were to them in the 1940s: threaths they cannot single-handely defeat. Much like during the Second World War, where air cover was required when operating in range of enemy air power, battleships of the 50s need escorts and cover to protect them from missiles. I don't think that makes them completely obsolete, one could argue that aircraft carriers requires the same kind of protection to survive in hostile waters. As long as the battleships are able to accomplish their mission (be it a ground-pounding operation or dealing a bloody nose to the Sverdlov swarm attacking their carrier group) they are still relevant in the grand scheme of war. Of course as long as they can be afforded. However when we reach the end of the 60s and the 70s, things are changing rapidly. Like some have pointed out, while small anti-ship missiles couldn't sink a modern battleships, saturating attacks or the use of the truly massive Soviets supersonic weapons are by now a threath so great, the battleship might very well be in danger of being unable to accomplish its task. This is where I think true obsolescence begins, when a weapon system is compromised to a point where it cannot be safely relied upon even when surrounded by the good amount of support. Now I wouldn't be able to tell exactly when it became the case but I think its safe to say that in OTL in the 80s, no one outside the US Navy could afford to operate battleships. By then they had such a niche role on the modern battlefield that taxpayers money could probably be funeled into other projects and nobody would be arguing about it.
IMHO, short of some massive POD, the most plausible way to see more battleship operated in the 1980s (and prior to that) is to have the Soviets build some. I'm not saying it would be particulary brillant from the Soviet Navy, far from it in fact. But let's say they keep Giulio Cesare in service (without catastrophic explosions), learning in the process how to operate a battleship. Stalin then push its big gun babies a little harder down the throath of the Soviet admiralty. Now its the early/mid-1950s and the Russians have 4-5 battleships/battlecruisers in service, they might as well put them to use (i.e let's hope they simply don't scrap them as soon Stalin dies) aren't they? With this threath floating around, I could certainly see Vanguard kept in service, maybe alongside a pair of the last two KGVs so that at least one BB is ready to put to sea at any time. I don't think the US would find it useful to keep in service more than the Iowas, but probably they would keep more in service at the same time and have some South Dakotas in reserve for longer.
So more battleships in the 1980? You won't get a dozen more but with the good incentive and the money, that's not impossible.
And honestly, who wouldn't love the sight of HMS Vanguard steaming alongside the carriers on their way to the Falklands in 1982?