Ok, I live in France atm.
SuperU, Intermarche, Carrfore are the Tesco's and Morrisons- There are also lots of Lidl's and Aldis. Our local Lidl is excellent, you will recognise many of the lines.
I use their w-up liquid and washing powder- it's the same as in the Uk.
Some of the French brands in the other supermarkets smell funny (to me anyway).
You really won't need to take all that stuff: mayo, ketchup, pasta, tins of veg, fresh milk, cereals (see Lidl's although hardly any are not sugary or chocolaty.)
What you can't get or only at cast expense in the"foreign" section are baked beans, tinned spaghetti, proper tea, cheddar cheese (vvv pricey), marmite, oxo, gravy, curry sauce/pastes, many jar type sauces. They also don't seem to have pasties, sausage rolls or pork pie types things for lunches.
I find meat expensive compared to the Uk, so buy the bigger packs which are on "promo".
No need to fill the car with basics, imho.