If the Franco-British Union happens, it will be as a war time creation, and I suspect that the other occupied European countries will be tempted to get involved in some form - I can certainly see complete military integration.
Honestly though, as soon as the war ends, I think the union would be dissolved, although it would result in a much closer Franco-British alliance with ready made common institions. I made a TL on this a few years back.
I will explore this in greater details in the subsequent chapters of my TL (I know I really need to get it restarted) but what is likely to happen during the war is the fact that if working separately France and Britain would easily fall prey to influence from either the Americans or the Soviets. This coupled with fancy ideas about what postwar Europe should look like, would provide an ideal breeding ground for ideals about a continuing Franco-British Union post-war, among at least some politicians.
Now of course this would lead to tensions since part of the political spectrum of both countries would be opposed to a continuing Union. The resulting Union government would also be a huge mess due to the sheer number of compromises required, but it could be reformed into something workable down the line after a decade or two.
There is actually a fair number of ideals and values on which the Union could be established. A form of "mild imperialism" could be one, with a desire to use the Union as a vehicle through which France and Britain prestige could be maintained and even expanded. Technological progress could be another one, especially if some politicians tie this to the legacy of France and Britain as "inventors of the modern world" and such.