The rivalry certainly precedes the age of nationalism (i.e. post French revolution), but from my reading is mostly due to the protestant-catholic and imperial rivalry in the early modern period, rather than being an important feature of medieval Europe.

There are signs of antagonism earlier. Edward I of England warned in the 13th century that the King of France intended to invade England and wipe out the English language (Bartlett 1996: 130), and in the hundred years war Joan of Arc famously wrote 'You, men of England, who have no right to this Kingdom of France, the King of Heaven orders and notifies you through me, Joan the Maiden, to leave your fortresses and go back to your own country'. However these conflicts were fundamentally dynastic, while nationalist sentiments were used by leaders they were not the crux of the conflict.

Much more important was the results of the reformation and then imperial rivalry (and the rivalry between English liberty and French absolutism). France became the leading Catholic power in Europe, so protestant England naturally felt a rivalry. Then in the late 17th and 18th centuries, France and England were on opposite sides of several wars (9 years war, war of the spanish succession, war of the Austrian succession, 7 years war). Within Europe, British policy aimed to stop French attempted hegemony, and outside of Europe, to capture French imperial possessions, e.g. in North America and India. Here there was certainly an important rivalry, Liah Greenfeld in 'Nationalism: 5 roads to modernity' writes about the Franco-British rivalry in the 18th century that led to the development of French nationalism. There's a good quote in Shovlin (2010) that I like that illustrates this:

"There are wars in which the nation only takes an interest because of its submission to its prince’, a correspondent of the Année littéraire noted in 1756, but ‘this one is of another nature; it is the English nation which, by unanimous accord, attacks our nation to take from it a good that belongs to each of us"

References

Bartlett, R. (1996). Language and Ethnicity in Medieval Europe. in Hutchinson, J. and Smith, A., Ethnicity. Oxford: Oxford University Press.

Shovlin, J. (2010). Selling American Empire on the Eve of the Seven Years War: The French Propaganda Campaign of 1755-1756. Past & Present, 206(1), pp.121-149.