How are you going to ensure indefinite leave to remain if you have only been there a few months by the end of the year - does one of you expect to be employed? It would be a great shame to move the kids and not be allowed to stay!
On the second language acquisition by immersion for a ten year old I would just say that although it's doable expect to work at it. Do not assume that your children will be fine - they might not be on their own, they might need help (even the seven year old) and they may well fall behind academically and need to repeat a year or do a year again. They may be very unhappy for many months. It's not nothing and children are not always endlessly flexible.
I taught secondary school children in England who took considerably more than a year to start to work at anything like their level after arriving without much English, and a Portuguese boy who joined year seven who developed selective mutism and was still communicating by drawing by year nine (though he fitted in well socially and had lots of friends despite not speaking - he was a talented artist and sports player).
An American girl left my youngest's German primary school four months after joining year four because she was deeply unhappy and understood nothing - she was the only child in the school who didn't speak German. Youngest was only in year two (and a boy) and she wasn't interested when he told her he was English, but I can't totally guarantee he did the cognitive gymnastics and actually spoke English to her as he often didn't really register which language he was speaking at that age, and it's really hard for bilingual children to speak the minority language out of context with a new person when surrounded by speakers of the majority language! The girl was moved to a private international school taught in English.
I know that's doom and gloom, I'm sure you'll also get replies from people assuring you your children will be fluent in six months, but I just wanted to point out that this move would be asking a collosal amount of your children - obviously you don't have a time machine, but you've left it seven years too late to be easy IMO!
Do you or your husband speak French or Spanish? I also knew families with two English monolingual parents who's children were born here who still ended up moving their children out of state school to private English medium school because the children's German wasn't good enough to reach their academic potential! Those families lived somewhat in an expat bubble and the children only spoke English in class and had English speaking friends and social lives, and being born in the country wasn't enough. In one of the cases there were also processing issues not diagnosed until late primary age exacerbating the language issue. Not all outgoing children have a natural talent for languages!
I'm not saying it's a bad idea - just to go into it with your eyes very wide open, your sleeves rolled up, boxes of tissues and chocolate at the ready and do absolutely loads of research - it's crucial you chose schools with experience of supporting immigrants and that you are prepared for either or both of your children to struggle and need massive amounts of emotional and practical support.
Add into that the fact moving for the medium or long term is a massive emotional rollercoaster for adults too - you'll question and second guess everything and even your sense of identity takes a hit unless you're already fluent in the language and can slot right into the community (even if you are fluent this takes work, as people who move to Australia or Canada or the USA will usually also attest).
Good luck - but weigh up the pros and cons carefully.
Don't forget that you'll need to stay at least 3 or 4 years and continue to expose your children to the language after your return to the UK for it to "stick" if your move is temporary - and on the other hand that if it goes well you may have a huge problem if your eldest is 14+ and settled in France and you want or need to return to the UK!