In Belgium this wouldn't be the case.

A king would be named King X until there is a King X the 2nd.

For example, King Albert the 1st was called King Albert until King Albert the 2nd became king in 1993.

I suppose it would be the same in most countries.

Any specific countries you're interested in?

As far as I can tell from a cursory search of Google Books, he was only referred to as "King George" during his reign, bar I think a single Hansard record? This fits in with convention in other places: it was always Queen Elizabeth, not Elizabeth II until 1952; nowadays we don't refer to Pope Francis I, and so forth.