There's a double taxation agreement, you will not be taxed twice.
Also not true that you have to be tax resident in France for school/healthcare. You just have to be resident.
More than six months a year in country, resident or not, you fill in a tax return.
School system is excellent, at that age they'll pick it up quickly. It's harder for you but you'll learn enough to get by well and if you stay a long time or really throw yourself into it, you'll get to very capable.
Pound is weak at moment against euro, so your currency power is down.
I'd recommend renting first (I really wanted to buy immediately), you get a feel for a place and areas, then can make a much better decision re buying somewhere. If I had bought straight away I dread to think what mistakes I would have made/opportunities I would have missed. Plus buying once you have some experience of the language and the system is much more sensible.
If you want to, just to be safe, move before certain dates too and get set up with a place to live and the paperwork for it, or your freedom to just move here may change.
I think it's a fantastic idea.
I would say that Scotland is a wonderful place too, people, culture, country, all very different to rest of UK, lovely property and may be getting ahead with kids age, but free university too.