4 years later I ... Contact Bike Register...
I'm guessing the bike's rightful owner is probably thinking it's a bit suspicious it's taken you 4 years to get in touch with Bike Register.
Ianal but I imagine your within your rights to remove parts you have bought for the bike but there may be a requirement for you to return the bike in the condition it was bought in? Ie the tyres may have been worn but there were at least tyres on it.
Citizens Advice website says "If you have spent money on improving or repairing the stolen goods, you may be able to claim compensation for the value of these improvements or repairs."
I don't know if there is a period of time after which stolen goods are no longer considered 'stolen good'? I was going to say there might be a limit on how long after a crime you can report it but we're always hearing about cases regarding historical sexual assault/abuse so that'd suggest not.
I think bottom line is that as far as you can tell, it's a stolen bike and now that you aware of that, it needs to be returned.