I am French and have been living in the UK for a long time with my British husband.
We have adopted twin boys (full adoption) 20 years ago when they were 7 months old (now 21).
Do they automatically take the French nationality?
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Automatic? No. Especially since you were living abroad. But your children may have the right to ask for it. This page says, among other things, that the France residency requirement is waived if the French parent doesn't reside in France. On the other hand, it mentions that an exequatur is required for the adoption to be recognized in France, if it was done abroad. Which I suppose is your case: if you adopted the children in the UK, you'll need to have this adoption legally recognized at the French Consulate.
Once this is done, since they're adults, they will have to talk to the French Consulate. They're the only ones now who can ask for the French citizenship.