Assuming a same-day connection from First, I've not had that happen before.

Indeed, I've booked cabanas when connecting to a European flight from First, sometimes via YouFirst, sometimes via Quintessentially. Neither one has so much as blinked.

If you're stopping over in London for whatever reason, then it would be correct that you would not have Concorde Room access for your onward European flight.

When you say you called the "concierge line," was that Quintessentially, YouFirst, perhaps your Executive Club contact, or someone you reached via reservations?