After giving it some thought, I think I have the right system. After the invasion, the UK monarchy is no longer hereditary. Instead, the French and Irish parliaments, in a joint session, elect the British monarchs.

Everyone's happy. Ireland and France have their democracies. England has its monarchy.

We can even draft a constitution with bounds on who can become monarchs. For example, they might need to be citizens of former British colonies. That might help keep the former colonies happy. Keep in mind both Ireland and most of France were, at one point, colonized by Britain (for France: See Hundred Years War, The Black Prince, and Joan d'Arc).