Fortunately for our little discussion, Britain's strategical position after a French defeat is not a matter of what-if history. There is a concrete historical example: Britain after 25 Juin 1940.
Britain's position after the fall of France was 'clinging on for dear life'. Oversees possessions were falling like domino blocks. Even at the height of the Japanese threat the UK insisted on Australian reinforcements for Europe. An incapacitated, impotent Empire, until either the Russians or Americans could be persuaded into an alliance.
That was the actual strategic position of Britain, in the event of a German conquest of Western Europe. It would not have been much different in the first decades of the 20th century. There was no such thing as blissfully remaining aloof, remaining outside of European affairs for Britain. Britain was a European affair. The entire Empire was build on Britain´s careful maintenance of a European balance of power. It is a British myth that Britain had (or has) the luxury of staying aloof of 'European' affairs.
There is great strenght in the isle of Britain, and a phenomenal strategic position. Britain is not easily under acute threat. The fantastically overstretched Empire, however, always was. In the event of a French defeat, the Empire had to be given up. Yes. This is what was at stake. France defended the British Empire. (As much as the more easily visible reverse: Britain defending France and her Empire). As it is, even the victory in WWI already meant Britain had to provide for many home rule / independence movements throughout the Empire.
A French defeat in WWI or WWII would, by and large, have spelled the end of the British Empire. As it did in actual fact in WWII. A French defeat means British concessions. To several demanding parties: independence movements (India), US pressure (an end to colonialism as the price for American intervention), imperialist competitors - see Japan in the Far East in WWII.
Likewise, a French defeat in either WWI or WWII would mean Eastern Europe will fall to the victor of the German-Russian struggle for supremacy in that part of the world.
Not that the world revolved around France. It is an interconnected world, that is the point. A British defeat would've meant the end of France as the world knew it, period.
Then again, perhaps France and Germany could've formed an alliance, close economic cooperation, and challenge the anglosaxons together...
Nah...that would never happen...
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