One day before I am supposed to leave, UKVI have returned my passports.
They included a letter saying that they will pause the application until I return those passports within 28 days.
I'm going to travel on my new second passport, carrying my old expired passport that has the ILR stamp. When I enter the UK, they will stamp that passport with a stamp that says they saw the ILR visa in second passport. I'm hoping that after I return the two passports back to UKVI, I'll be able to go to France on my second passport and it will be fairly legit with that stamp. Once in the past we forgot to bring the passport with the ILR and the Border Officer told us that it was OK since there were ohter stamps that said I had a visa in the second passport.
Drama over.
Thanks everyone for support.