One of the interesting quirks of losing the empire is that there are a lot of precedents for 'Westminster model' countries becoming republics. The straightforward way is to give the reserve powers of the monarch to a mostly-ceremonial President on the Irish or Israeli model, and vest the rest formally in the government (which coincidentally also makes them subject to more parliamentary oversight). In the case of Canada/Australia etc. the Governors-General are already performing such a ceremonial presidency in reality. All that's needed is a process for electing new ones; fairly straightforward.

You're right that it would require international cooperation, though: the British parliament doesn't legislate for the other Commonwealth Realms any more.