BA absolutely do downgrade people on paid tickets. DXB tried - and failed - to do so with me and I've seen a colleague downgraded from F at LHR check in (before being moved back up at the gate). CX do the same, I've seen a colleague go from J to Y ex SIN to HKG on a full fare ticket.

To the OP - it happens rarely and even less so now given the poor F/J loads and rampant overselling of Y. Upgrades are far, far more likely.