Just to say clearly before I say anything else. DS is now 21, he was born in France (Paris), we left when he was 4 1/2 to live in Rome. He had one year in a bilingual nursery in Paris. In Rome he had two years of bilingual school in the morning, with Italian school on the afternoon, supported by private tutoring in Italian. Then 4 years in a bilingual school with our final year there in an Italian school. We came back to France (Paris) when he was 10, he spent two years in a semi bilingual school in Paris then we moved to the south of France where he was in French school until he was 18. I wrote everything below but then realised a bit of clarification at the beginning might be helpful.. I have met an enormous amount of parents who have emigrated and were putting their children through a foreign/bilingual education system.
Education is the most difficult one but certainly not insurmountable. It depends where you will be living to a certain extent and the school choices available. I know a lot of English speaking children who have had terrible struggles with the language (in France and Italy) because the grammar is so difficult in the beginning. Some of them to the extent that families have left after a year of distress for the children. It doesn't have to be like that but it needs careful thought and planning. Many people make the mistake that the kids just 'pick it up' , they don't, especially at the ages of your children. If they have had a monolingual life before it can be very difficult for them.
If you are in an area that has bilingual schools choose one if available that has an immersion class designed to integrate children into the French language and school system for the first year. Supplement this with private tutoring. After the first year either switch to a French school or carry on at the same school if it is mainly French after immersion year. Then consider a French school. DS arrived back in France at 10 years old, did one year immersion class, then a second year in the same school where he was mainly taught in French but had supplementary lessons in French for foreign language students. He then changed to a French school until he finished at 18. If no such classes are available you absolutely have to get them private tutors.
Most French schools are very welcoming to foreign students and will help as much as they can but it will obviously be difficult for your DC..Look at private schools as well. The bilingual ones are more expensive, but it may be worth it for a couple of years. French speaking private schools are mainly catholic (but with no religious lessons), very and are around 2-3k per year depending on the age of the child.
You and your DH won't be able to help your children much but all of the foreign families I have known if their parents make a real effort to learn French the children tend to do better with the language. DS's teachers in France and Italy have all said this as well.
DS didn't have extra tutoring on returning to France but that was because he already spoke some French and he spoke Italian fluently. In Italy he had had about 10 hours of school every week devoted to grammar for 6 years and it was close enough to French he didn't struggle with the grammar.
If this is a long term move you really have to think about long term education. Your 10 year old will have to take the Brevet at 14 if he/she is in the French system. That's a lot of learning to catch up on in 4 years. It's not impossible but it is difficult. The Brevet is similar to O levels (if they are still called that 😊), the results will then determine where they will go to Lycee. Either general or technical . That in turn determines future careers. You know your DC and what outcome is likely for them at that age.
Post 18 education is a whole other nightmare. If your DC have EU passports it will be much easier for them. They will not be eligible for subsided education in the UK despite being British citizens. You will have to pay for them as overseas students. So they stay in France but then you enter into the world of prepa and Grande Ecole if they are academic and their French is good enough. The alternative may be an English speaking university in Europe but they are costly. If your children are really integrated by then they will probably want to stay in the French system because in the the last year of Lycee all the talk is about where they are going, their choices for post 18 education (which is really complicated, not just what university to pick).
Sorry for such a long post. But it is so important. It's a minefield we have had to pick through for 20 years with DS. DH and I spoke, sometimes on the verge of arguing about education for years. He favoured long term bilingual school. I didn't as I knew that the level of both languages was very poor. With your children they could spend their education in France in an English speaking school or a bilingual school. But what's the point of that? Both languages will be poorer for it. You are in France and giving them the opportunity to become bilingual is a great gift.
My DS is now studying for his Masters in engineering at one of the most prestigious Grande Ecole in France. I have a Polish friend who I met on a basic French language course during our first month in France 22 years ago. Her daughter is 4 weeks younger than DS. She is also now studying engineering at an equally prestigious school. The pair of them turned down places at the Sorbonne to be in their schools. They are both bilingual and have a great future to look forward to. I'm not saying that to try and brag, I'm saying it so that you can hopefully see despite the difficulties it can be a wonderful thing for your DC.