Thanks to JAA-standardisation, there is no reason why you can't, in principle, get a job in England or another JAA country with a French licence. During the golden days of cadet sponsorship, even BA selected into its Trainee Pilot Programme quite a few Europeans whose first language isn't English.
But herein lies the key - if you attend an interview with an English airline, your language skills must obviously be at a high level - you need to fit in the organisation.
You didn't say at what stage of your training you're at, but if you are doing the ATPL theory now, a good idea might be to go to a UK FTO (training provider). You'd get used to flying to UK procedures, and brush up on your language skills as well.
Don't forget, with a JAA licence, there are many other options open to you (in addition to France and the UK).
Good luck,
KK