I moved to England in primary school and after about a year my English was OK. Not bilingual by any means, and I won’t say “fluent” as clearly that means different things to different people, but good enough for any conversation and good enough to keep up at school. (For adults, the process in my experience is much slower, not least because it is harder to achieve the same level of exposure – you will not be surrounded by French speakers all day.)
But yes, kids will in time pick up the language and catch up with peers academically. However, emotionally, I found the first year was very tough: adapting to a new culture, a different education system, making new friends, etc, etc, all alongside learning the language. It is not something I would want to put my kids through unless I was planning to stay in that country for a few years as you would just end up enduring the slog and the upheaval and miss out on the eventual rewards.
Ultimately, though, I think the biggest issue with your plans is finances. Commuting to a different country is expensive (flights, accommodation) and will hinder your efforts to really settle in France. The housing market there does not have a reputation for steady price rises, so if you sell in the UK you may struggle to buy in the UK again in the future should you wish to move back. (Good schools also fill up so kids may not be able to go back to the one they were at.) After a year (and perhaps even years), your French is unlikely to be good enough for a proper job in a French company. There are, of course, some professional jobs in multi-national companies where language will be less of an issue, but how realistic it would be to get such a job would depend on your CV, where you are in France, etc.
You need a firm, realistic plan for work, housing and finances, and a detailed budget to be sure that it could all work. Otherwise you risk exposing the whole family to a lot of heartache with no certainty of being then able to return to where you left off if it does not work out. And I offer this note of caution as someone who has lived in a number of countries, is generally up for adventure and values multi-cultural and multi-lingual experiences.