Obviously, this is a counter-factual question - we don't know for certain what the conditions would be or how they would have played out in practice.

However, we can examine the 'constitution' of the British Empire and see what the constitutional-legal position would be.

The first point to note is that the quote marks around the world 'constitution' are relevant. Some parts of the Empire did have recognisably written, codified constitutions of their own, but the British Empire as a whole never had a simple or unified constitution. The 'British constitution' was (and remains) a hotch-potch of statutes, conventions, norms, practices, judicial decisions and vague notions, and that 'make it up as you go along and call it flexible' approach extended into the relationships between the different parts of the Empire.

Second point: it probably isn't accurate to think of 'Britain' having an empire in the first half of the 20th century; the United Kingdom, which was recognised as consisting of distinct nations - England, Scotland, Wales, and the Northern rump of Ireland - was part of the Empire; the Empire as a whole, rather than the United Kingdom, was the focus of loyalty and patriotism. That's important to your question, because losing the UK isn't necessarily like losing the 'core' or the 'head' of the Empire - it's losing a part of it, but the head - the Crown - can reappear elsewhere.

Third point: The Empire was a heterogenous collection of territories. It included, besides the United Kingdom, the 'Old Dominions' (Canada, Australia, New Zealand, South Africa, Irish Free State until 1937), India (which was not a Dominion, but together with Burma had a fairly large degree of internal self-government after 1935), the self-governing colonies (Southern Rhodesia, Ceylon, at various times Malta), Crown colonies with varying degrees of autonomy from little outposts under the direct rule of a military or naval governor, to those with fairly representative institutions. In addition, there were protectorates (which had their own 'native' governments under the tutelage of a British Resident, who 'advised' them) and mandated territories.

Fourth point: The Dominions had a special status. From 1926, they had been recognised by the Imperial Conference (the meeting of the Prime Ministers of the Dominions) as 'autonomous Communities within the British Empire, equal in status, in no way subordinate one to another in any aspect of their domestic or external affairs, though united by a common allegiance to the Crown, and freely associated as members of the British Commonwealth of Nations.' This had been put into law by the Statute of Westminster of 1932, which essentially, except for one or two specific purposes, gave the Dominions 99% of the substance of independence. The 'Commonwealth' was to run on mutual cooperation and ties of culture and loyalty amongst its members, not on top-down direction.

[There's a great story, illustrating this, from 1941 when the Japanese invaded Burma: British Prime Minister Winston Churchill wanted to send some Australian ships to the area to reinforce the defence of Rangoon, but the Australian Prime Minister, more worried about the Japanese threat to Australia's own territory than the loss of Burma, told Churchill to stuff it. That's how the 'Empire and Commonwealth' worked at the height of WW2: Britain could not, even for the defence of the Empire, order the Dominions around. The relationship between them was horizontal, not vertical.]

So, with all this in mind, the loss of the UK would not have greatly disturbed the Dominions. The Commonwealth, and its point of unity - the monarchy - could still have continued without too much trouble.

However: I said that the Dominions were '99%' independent. The other 1% was that in Canada, the constitution could only be changed by the 'Imperial Parliament' (as the Westminster Parliament was called) - because there was no agreement between the Canadian federal government and the provinces as to what the amending formula should be (this wasn't fixed until 1982). A work-around to this would have to have been found; not doubt, if the UK were conquered, the Canadian Parliament and provinces would, sooner or later, have found a way of working it out. A way would most likely have been found in law to give effect to whatever political agreement was reached: basically, 'patriation' of the constitution unilaterally.

What about the 'non-dominions', the Crown Colonies? Well, again, technically they were subject not to an entity called 'The United Kingdom', but to 'The Crown in Right of the United Kingdom'. Even if the territory of the United Kingdom were occupied, and even if the King-Emperor had to flee to Canada (there were actually contingency plans for that), the continuity of The Crown in Right of the United Kingdom would still be maintained, so (following Point 2, above) the colonies would still have been parts of the Empire. The Imperial Parliament had the sovereign power to legislate for the Crown Colonies, but it was not necessary to do so: the Crown could legislate form them by Order-in-Council, either exercising those powers directly or conferring them upon the legislative council of the colony.

Any action that the King needed to take in relation to the Empire or its colonies on the binding advice of Ministers could either be taken on the advice of a UK Government-in-Exile (assuming that the war as a whole was not lost, and that the intention was to recapture the UK and return to it), or, if that eventually proved impossible, it would not be beyond their wit to devise a legal way of transferring that authority to the Canadian Privy Council.

That's my assessment of the legal-constitutional situation: basically, the British Commonwealth and Empire was bigger than the United Kingdom, and would have continued - almost unaffected - if the United Kingdom, as a mere part of the Empire, happened to be lost.

However, this does not account for the military, economic, practical, psychological or political effect of the UK being invaded and occupied. No doubt, if that happened, it would be an enormous blow to morale and to war-fighting capacity. Confidence in the Empire and in the war effort might have collapsed in some places; perhaps leaders in colonial legislatures might have felt it was a good time to push for independence; perhaps regiments raised in the colonies, and used both for internal security and war-fighting, might have mutinies. I don't know. That's getting too much into speculative matters and I don't think it is possible to weigh and consider all the factors without a very well-developed, detailed hypothetical scenario.

The point is, if the loss of the United Kingdom caused the Colonies to become independent, or to disintegrate into chaos, it would have been a political shift caused by loss of confidence, morale, resources, or military capacity, not an automatic operation of law (because they were not dependent on the United Kingdom, they were fellow parts of an heterogenous Commonwealth and Empire hanging on The Crown, not on the United Kingdom as a state).

I'm tired, I'm not writing well. But that's the essence of the answer as I see it.