Is only the backrest wobbly, or the underside also? If it's the underside, then it could be down to loose seat rails (easy to fix). But, if it's the backrest only, then I have no idea. My backrests are also a bit wobbly, but I haven't stripped them down to see what's happening or if it is even possible to repair them.
How do you know that it is not the rails? I am guessing that the screws fixing the rails to the underside of the seat are loose. To remedy it, you need to remove the whole seat, which is not hard to do.
A common fault is that a small carriage on one of the rails doesn't lock in position, It happened on my 16vt but didn't look into a complete remedy - however, fully pushing the seat back or forward (can't remember!) usually re-engages the carriage.