There were a number of notions in the 1960s for a Amphibious Fire Support Ship, or LFS.
It is out of some of that work that the OTL Mk 71 developed.
Earlier thoughts were for a ship with 1-2 new 8" guns, a number of the new rapid fire 5" twin mounts (never entered service), some rocket launchers a la Carronade and various missiles and rockets, including one called Sea Lance.
There was some tentative exploration of a 12" weapon. I'll have to dig out my copy of Friedman's US Amphibious Ships and Craft if anyone is interested on more detail. IIRC, there are was an extensive chapter regarding the LFS development.
However, even that failed due to cost, lack of an immediate need, doctrinal issues and the continued presence of WW2 ships.
In terms of any new battleships, there is unfortunately no chance of any new construction - they simply cost too much, took too many men, didn't bring anything really necessary to the force equation, were extremely vulnerable against missiles, subs and aircraft and were viewed as obsolete in the face of the capabilities aircraft and missiles to do the job they once did.
It would be far better to channel any funds from battleships into develop of a 175mm or 8" gun capable of being deployed on half a dozen LFS type vessels, along with the DX/Spruance class.
8" could go fairly far inland, particularly with new types of ammunition being developed and tested under Project Gunfighter and other goings on at the time.
Now, in the 1950s, given a lot more money, the BBG/BG conversions may go a little further, but still fail. Problem there is Polaris taking a lot of funds and that most of the more useful conversion plans resulted in the ships losing their main gun armament, which is all that sets them aside from a missile cruiser, really.
Even then, economies of scale kick in - one can get a lot more DLGs that can be in a lot more places than one BBG, and they can basically carry all the necessary SAMs.
Further back now, to the 1940s. Have Stalin be a little less indecisive about his Ocean Going Navy plan, and have him actually get down at least 4 Pr.24s, along with a number of Pr.82s (later modified into 82Rs). Add in a few Pr.66s and there might, just might, be a chance for battleships sticking around a bit longer; have him live to 1955/56 to see it's completion.
There is a window of opportunity when the necessary all-weather attack aircraft could not operate off carriers in all locations, so that battleships may still have a role 'way up North'. But it is a very small window.
Balanced against that possibility are a fair few things, though: The Soviets being able to afford the massive expense of a fleet programme, along with all other costs; the most likely Western response is simply to keep a few more carriers about, given their huge superiority there; all of the major research and development was not going into guns, but into aircraft launched weapons capable of sinking any ships - in the USSR, US and Britain.
Another decade back, to the 1930s. Changing how WW2 works out could extend the life of the battleship for a little while, but that involves a lot of fiddling and points of departure. Reduce the impact of aircraft, of carriers, delay development of guided bombs and heavily delay the development of atomic weapons. All of those factors play a part in killing off battleships in the late 40s and 50s.
There is the influence of the 1920s, with the impact of the Washington Treaty and the 'freezing' of development of the ships for quite some time. In some cases, the capabilities for efficient construction, gun pits and armour manufacture took quite a beating.
Having them perform as mooted in the Great War could help them a little, at least in starting butterflies.
Finally, going back to the turn of the century and the Belle Epoque, if we delay the development of aircraft, it can add a few more years onto the capital ship.
But to have a realistic situation where battleships remain in service, we need a combination of almost all of these factors; the battleship was already facing the harbinger of it's destruction in it's seeming heyday of 1900. I haven't been able to figure out an effective post-1900 without adding ASBs with a steampunk fetish; I finally gave up on that POD and started afresh with an entirely new world.
On a few other points, nuclear powerplants are not the best idea. This is a ship going to go into danger, close in to shore. The armoured belt is not in the best place to stop modern missiles (and by modern we mean post 1950, when the first Soviet shaped charge AShMs start to appear, with one of their purposes being killing Western BBs) or modern torpedoes.
The design and organization of the ship isn't optimum compared to more modern ships in terms of shock resistance.
However, the main point here is that putting a nuclear reactor or four in a place where people are going to be shooting at it and bombing it is not the most sensible of options.
Furthermore, cutting up the ship to gut out the powerplant is going to involve a lot of work, expense and damage to the armour scheme.
Reducing the crew carries with it issues of damage control, maintenance of systems and the smooth functioning of the ship. Some reductions may be possible.
Finally, the speeds are a bit optimistic as others have said. Check out this discussion for more on that issue:
http://www.navweaps.com/index_tech/tech-028.htm