(1) Most Mediterranean areas of France and Italy have similar or rather higher rainfall than SE England, contrary to popular belief (Marseille's annual rainfall is 546mm, about the same as London, but hilly topography means that nearby inland locations are generally rather wetter. More typical is Naples, where annual rainfall is 915mm, similar to Manchester.)
(2) Many parts of SE France benefit from large amounts of water coming off the Alps and passing through their area in large rivers such as the Rhone, which they can harvest.
(3) Expecting a long dry summer, and having lower population density than SE England, many Mediterranean areas of Europe devote considerably larger land areas to storage reservoirs than we do.
(4) Areas of SE France without convenient Alpine water (eg, around Nice) do have very serious water shortage problems in some years. Severe restrictions on the use of water may be imposed in such cases. In some years they suffer the opposite problem of torrential summer downpours which make the Boscastle flood look like a slight inconvenience.
(5) Water in SE France is universally metered. Prices per m3 of water in SE France are generally rather higher than in Britain. In some areas of known water shortage, eg around Nice, you have to "reserve" your water use in advance and pay for that reserved quantity without any refund for under-use. If you exceed your reservation, there is a much higher penalty rate/m3 you must pay. This means French municipalities (which often employ private water companies) have a lot more money to spend on making sure they have supply than UK water companies.