Losing BA Silver after picking Club World seats… do you keep them?

From what you’ve described, yes.

If you selected your seats while you still had Silver and the bookings are already ticketed, those seat assignments should stay as they are even after your status drops to Bronze.

BA doesn’t usually go back and reprice or remove seat selections just because your tier changes later. The benefit applied at the time you made the selection.

The only time you’d potentially lose them is if there’s an aircraft swap or schedule change that forces a reseat. Otherwise, you should be fine.