The most obvious POD for a Communist Britain is the General Strike of 1926. If the government failed to adequately prepare (unlike OTL), if the response was more heavy-handed and violent, and if the King did not adopt a conciliatory tone, there could have been an attempted overthrow of the government. It's unlikely - there were significant revolutionary elements among the strikers, but revolution was not their declared objective.
Nevertheless, if they succeeded, then the government and royal family would flee abroad. Most likely to Canada, but if some of the colonies can be kept together they would still administer these themselves. George V refuses to leave, not willing to abandon his subjects to socialist tyranny. He suffers the same fate as his cousin, Tsar Nicholas. Edward VIII becomes King, and is crowned in Ottawa.
Meanwhile, as decolonisation accelerates (probably earlier for a 'British' Empire without the metropole) and the legitimate British government feels the need to leave Canada as the constitutional links are slowly severed, a new headquarters for Parliament is established on Bermuda. All remaining Crown Colonies willing to remain British (probably most of today's Overseas Territories, possible excluding the Chagos Islands and the Sovereign Base Areas on Cyprus due to British weakness) are united into one 'nation' with their own Parliamentary constituencies, electing MPs to the 'British' House of Commons in Hamilton, Bermuda. The King shares his time between Canada, Bermuda, and the other Commonwealth Realms.
The Channel Islands, the Isle of Man and the Shetlands fall under Communist control, but I suspect that Gibraltar would remain legitimist (or possibly be taken by Spain, depending on how much military power the legitimate government can muster).
