Hi ! We moved from London to Madrid two years ago with similar-aged kids so happy to share what we found.
On schools — the Lycée is obviously great but the waiting lists are brutal. A few alternatives worth looking into:
https://wisdomschool.es — this one really stood out to us. It's a state-subsidised bilingual school (concertado) with around 60% of lessons in English, native-speaking teachers, and they're an official Cambridge exam centre. The bilingual setup is genuinely strong, not just a label. Worth contacting their admissions team directly (they have an English-speaking contact which helps a lot when you're doing everything remotely).
https://www.sek.es/l/colegio-internacional/madrid/ciudalcampo/ — fully international, IB curriculum, very used to welcoming kids mid-year which could suit your January move.
Colegio Alameda de Osuna — strong bilingual programme, good reputation among expat families.
The British Council School — obvious choice for English-speaking families, though pricey.
On the French side — given your kids already speak French, the bilingual concertados might actually serve them better value-wise than full international fees, since they'd already be ahead on the language side.
For life in Madrid — ChamartÃn and the north of the city (Hortaleza, Las Tablas, Sanchinarro) tend to be popular with families moving from the UK. Good transport links, newer housing, calmer pace than central Madrid.
Feel free to DM if you want more specifics — happy to help!