It is likely that had it happened, the monarch of the United Kingdom would've become the head of state. There hasn't been a King of Great Britain since the 1st of January 1801. The proposal accepted by the British War Cabinet on the 16th of June 1940 says

France and Great Britain shall no longer be two nations, but one Franco-British Union. The constitution of the Union will provide for joint organs of defence, foreign, financial and economic policies. Every citizen of France will enjoy immediately citizenship of Great Britain, every British subject will become a citizen of France.

This implicitly supports a united head of state, and since it is incredibly unlikely that Churchill and Chamberlain were republicans or prepared to subordinate the King to a President that would mean that the monarch would rule jointly over the United Kingdom and France or that France would be absorbed into the UK as a home nation retaining distinctive legal and cultural customs like Scotland. It was never developed on, and the French establishment immediately rejected it and the propsal was suspected to be nothing more than an attempt by the British to claim French colonies as their own.

This answer is assuming that you meant King of France and not Kim of France when writing this, if I am mistaken I apologize — I have no knowledge of the Kims of France.