This is more or less what we did in the end, although it involved temporary accommodation in both the UK and France. A real pain with 2 young kids (youngest only 2 months at the time!).

It may seem an obvious solution, but trying to synchronize all the timings was a nightmare - completion on the sale with buyers agreement, finding a rental property in France etc - hence the 2 holiday cottages in UK and France to take up the slack.

However, 9 months on, we've bought a house, eldest is settled in school and all is going well, so it was worth the hassle

D.