Would it also help to expand the definition of Brittany and turn it into a Kingdom in its own right? A hundred years war that grinds down into a true stalemate with neither side able to accomplish much in each way. With no side able to make the victory decisive, and with land constantly changing hands, a treaty could, potentially, divide the lands of the two warring sides. The western half of Normandy, Maine, and Anjou could join the Duchy of Brittany in a new Kingdom of Brittany (or more likely continued to be held under the same crown) while the southern and eastern parts were integrated into France.
Perhaps, also, it would help if an independent power set up in (parts) of the Aquitaine and in Toulouse. It would act as a check on French expansion.
That way, you would have a large(r) presence on the continent, which is important enough to be held as an integral component to the nation, and perhaps a second crown. Expansion against Scotland would be even more inevitable, as France is an even greater threat against the union. With the Scots integrated either before or after independence, they would only have to worry about securing the isles and warding off the French. That might involve a more entangling alliance in Europe, and will cause a complete change in continental politics.
Might England even seek out the East Indies trade earlier in order to supplement their resources against the French? They could send an expedition to the west to try and find a shorter route. Could be interesting seeing this England becoming an alt-Spain (with Toulouse/Aquitaine/Navarra taking the place of Portugal?)
For it to be a United Kingdom of Great Britain, however, won't we need one side to inherit the crown of the other? Might it be more likely that England/Brittany inherits the other? Then you might have the United Kingdom of the Britains and Scotland instead. Be an interesting twist.