I was watching the most recent episode last night and was a bit confused by the jabs. I can understand the French chef feeling his cooking is superior, but then I think about the food served in "Downton Abbey" and it was anything but bland. Maybe that's just because they were English nobility? I always thought the rationing during the World Wars was the major cause for English cuisine being seen as "bland".
I have one point I wanted to add to my older response, which u/Broke22 linked below.
While France has influenced England's cooking for a long time (which we can see linguistically through our French-derived words like "beef" or "pork" to mean cow or pig meat used culinarily), this exchange got a kickstart after the French Revolution. With the end of the French nobility came also the end of countless French chef's jobs. These displaced chefs went on to found the first restaurants - a novelty for Englishmen and Americans traveling to Paris which spread quickly. They also traveled and often found work in the kitchens of nouveau riche industrialists in England and America, which I assume are the main characters of this TV show based on the title. Having a French chef was a status symbol whose existence in itself implies the inferiority of English/American food and added to the stereotype.
I also wanted to crosspost this comment by u/Bigglesworth_ which provides historical references for the existence of this stereotype before wartime rationing:
A. V. Kirwan's Host and Guest of 1864, for example, says that "The cookery of England is, with the greater part of the nation, an object, not of luxurious desire or morning meditation, but of plain necessity and solid comfort." Kirwan also quotes Voltaire saying "though (the English) have twenty-four religions, they have but one sauce", though that may well be apocryphal and appears to be variously attributed with different numbers of religions and sauces. By the 20th century Marcel Boulestin’s Simple French Cooking for English Homes of 1923 remains scathing; "I am also assured that many English cooks find cooking a tedious drudgery; perhaps that is because they have no imagination, or are not given a chance of showing initiative. It must certainly be extraordinarily dull to send up boiled potatoes and boiled cabbage meal after meal".... "Both wars brought about expediencies in the allocation of food but neither disrupted the long-term pattern of food consumption in Britain. Instead, if there is an overall pattern to the twentieth century from the point of view of food consumption, it should be to consider the period from the 1890s to the 1960s as one of stability – the final era of plain fare."
Edit: Also this other answer by the same user which directly addresses your point about rationing.
More can always be said, but take a look at these answers from /u/luiysia and /u/GrunkleCoffee