Hi, welcome to Just Answer. I will help you with your question.
If the other motorist has already agreed to the repairs, and effectively conceded liability, then you should have a pretty straight-forward task of issuing court proceedings to recover this money from him/her. You can issue court proceedings online, in the small claims system (for less than £5k), at www.moneyclaim.gov.uk
This is the cheapest and fastest way of doing it to recover money you're owed.
However, it's always sensible and reasonable to write to him insisting that he pay first, before issuing Court proceedings and incurring further costs and starting a formal process.
I hope this answers your question. If you need further information, just let me know. Please do remember to rate my answer as highly as you can.