Or, maybe, the Allies and Soviet Union won WW2, and both setup their system of government in the states they controlled and/or their former colonies as they became independent.

Then there was that whole cold war thing, and the USSR lost its client states, most of which adopted the political system of the winners.

The fact that so many democracies derive from the UK means that democracy is not a default state and to be successful needs centuries of tradition and institutions to make it work well.

I don't see how that follows at all. Democracy seems to be working out pretty well in Timor Leste, which doesn't have a long tradition, but not in Thailand, which does. Switzerland was never ruled by the UK and democracy works well, Zimbabwe was ruled by the UK and democracy hasn't worked out.

There's a lot more going on than some kind of simplistic explanation like this.