As Kate_Canuck alluded to, looking into flights from London directly to Nice also makes sense.
The Easyjet flight from CDG to NCE on the night of June 15th is currently €161. Easyjet flights from London (various airports, but not LHR) to NCE are between £64 and £79 for evening flights on the 15th, so it's a comprable price to what you'd be paying for the easyjet from CDG.