Well, firstly, worn tyres can make an Omega tramline quite badly.
Secondly, check all steering links that you can see - track rods both ends and steering idler.
Thirdly, I'd check the wishbone bushes are OK. They have a pretty short life so if they haven't been touched in a few years they are almost certainly past their best. They have to be very bad to fail an MOT here, but can cause the car to tramline badly long before that. Bottom ball joint is also a potential issue as is the top strut mounting, but they cause problems far less frequently.
Fourth, once everything above has been covered, get a proper geometry setup done and see how it drives then.
Fifth, if it's still not good, set the steering straight ahead, and, with the engine running, see how much "lost motion" there is between turning the steering wheel and the road wheel starting to move. If it's anywhere near the 75mm figure, you might be able to remove some of the play by adjusting the damper on the top of the steering box by moving it in tiny increments and re-checking play and self-centring to ensure it's not too tight. Over-do this and you can kill the steering box (then again, if you've proved it has unacceptable play, you might be happy with that risk).