A power may temporarily take control of a territory originally belonging to one of its allies (gaining its income and the use of any ICs and/or AA guns there) only when recapturing that territory from an enemy power while the original owner’s capital is held by the enemy. See page 20 of the Europe Rulebook. This has been true since Classic.
There is no exception made for French territories.
The “only” is not on page 20 of the rulebook. In fact, this situation isn’t addressed at all in the rulebook, and that is probably why we have a thread about it. To me, it seems more intuitive for the income to go to the UK.