Have you considered other countries as well? If tuition fees are an issue, Germany, Austria and Switzerland have low tuition fees for Master's (even for foreign students). Many programs are taught in English.
In terms of the process for the UK: I applied to several MSc programs in applied maths/stats in the UK at one point, in general you just need to send them a transcript, CV, 2 LORs, and some sort of personal statement. They don't care about the GRE nor do they have an equivalent. I don't know how they do things in France.
All universities should state their application deadlines somewhere. If these deadlines are less than ~ 6-8 months before the start of the program I would recommend checking whether international students can apply earlier, because you will need some time for the visa.
For UK MScs, at least in my experience, you can forget about financial aid if you are a foreign citizen. There are perhaps a handful of scholarships available.