A bit late for you this year but may come in useful in the future:
Normandy and Brittany has some good fly-fishing rivers otherwise head for the hills.
In the Limousin you can fly-fish in the Vienne and Creuse head waters between Limoges and Gueret near to Lac de Vassiviere. The Haute-Vienne has publised some useful information here: https://peche87.jimdo.com/nos-coins-de-p%C3%AAche/parcours-truites/
You should be able to find other similar information if you search on Google using keywords such as peche, truite, mouche which means fishing, trout and fly respectively. Add a department name or code number to narrow down the search area. Then use a translation service to change the information into English.
There are some beautiful areas to fly-fish in the Midi-Pyrenees in the rivers Lot, Dordogne and tributaries. But beware that some fly-fishing stretches are owned privately or by communes and may be private or subject to day ticket fees. It isn’t as free and easy as coarse fishing.
Fly-fishing waters are classed as Category 1 whereas coarse waters are Category 2. On maps published by fisheries departments you will see that the head waters are represented in a different colour. This differentiates the first and second categories. There maybe signs on road bridges or along the rivers defining the category and outlining any regulations.
The trout season ends in September (each department may have its own closing date) and re-opens on the second Saturday in March.