Well rhal96, may I ask you something, where do you want to do 1000 to 2000 hours on your EASA ticket? Probably in EASA land? Therefore you need the visa first.
But there is one thing my esteemed colleagues from that island that is now a third country forget, not everybody needs an ATPL. If you want to fly in the Alps and do mountain stuff, you need other skills. Mountain training, sling load. Many flight schools don't even bother with ATPL courses in EASA-land. By far most pilots go the CPL VFR-only route and get jobs. ATPL you need only, if you go for the big ships flying passengers (more than 19). For example the pilots that fly Super Pumas, Kamovs in France or Switzerland, none of them has an ATP (
AFAIK). And I pretty much doubt that many have IFR tickets. Ex mil does though. They don't need it, because all the ships are VFR only and have just a few lousy seats for the ground crew. My advise to you is, if you want to fly in the Alps, talk to a flight school in your region. Mont Blanc Helicopters for example. Do the PPL, do the CPL and if necessary IFR. And then you take any small freelance job you can get. Many operators actually prefer a pilot they have trained themself. If you stick with a company and do every stupid job they can think of, they might like you and give you a chance. The sweeping the floor route might still work and you can stay in France. And BTW, nobody will sponsor you in the Alps with an ATPL and no mountain experience. They have more than enough pilots to choose from. As a Brit, you have to apply for a new visa for every country you want to work in. You would need a really special skill. And if you don't loose your visa now, you can become a french citizen in a few years and all your troubles with visa are over.