Leave out the big five nations for a moment (UK, USA, France, Japan, Italy)
Minor battleship nations in 1921:
Argentina (2 dreadnoughts)
Brazil (2 dreadnoughts)
Chile (1 dreadnought)
Soviet Union (4 dreadnoughts, mostly inoperable)
Spain (3 dreadnoughts)
White Russia (1 dreadnought, soon to be interned in Bizerte)
Germany (6 predreadnoughts)
Greece (2 predreadnoughts)
Turkey (1 battlecruiser)
Sweden (3 coastal defence ships)
Australia (1 battlecruiser to be disposed)
Europe:
Spain and the Soviets will look to get newer ships in the 1930s, but will run into internal problems.
Dreadnoughts are well back on the list of things to acquire for Greece and Turkey, but if one get one at some stage down the line in a world where everything goes right for them, then the other may feel motivated to mirror that move.
Sweden could afford a smaller dreadnought theoretically, but it would be too large for their use and purposes in the Baltic, just as the Tre Kronor cruisers were post WW2.
Other than these nations, there were precious few who could afford to buy and maintain a battleship, with only the Netherlands coming to mind as a realistic operator.
Poland occasionally gets a mention in some neverwere ship compilations, but lacks a deepwater port or a definitive need, not to mention funding.
Yugoslavia, Romania and Bulgaria similarly lack the money and support, although they would be an interesting prestige ships in a very, very, very unlikely state of affairs.
Denmark, Norway, Albania, Belgium and Portugal are out.
There were a few Austrian predreadnoughts that were ceded to Britain, France and Italy that were scrapped in the 1920s that could be played with given the right (and twisted) motivation.
Outside Europe:
Canada and Australia could get into the battleship game in the interwar period, but are constrained by treaties, finance and internal politics. Fisher did recommend South Africa and New Zealand operate battlecruisers, but that is extremely unlikely in view of manning and economy.
A battleship as flagship of the Royal Indian Navy would not be impossible, but would need so many changes as to be close; funding could come from some princely states in a similar fashion to HMS Malaya.
Siam wouldn't be able to afford even a predreadnought in 1920, but did get two 3000t coastal defence ships in the late 30s; something to work with there.
China would theoretically be able to purchase a predreadnought if it was going cheap, but would need a lot of stability, funding and quite a few changes to do so.
Of the other South American nations, Peru is the only possible battleship operator, and did operate cruisers; I do remember reading somewhere about a possible sale of Vanguard that really was much ado about nothing.
Mexico is too broke after the Revolution, and Ecuador, Venezuela and Colombia would require a strange world indeed to operate battleships.