I had this situation years ago, albeit with a very small employer. I got advice from a friend of the family lawyer who talked me through what i needed to do for small claims court (not sure what the max claim is but you’re well under it) and I also spoke to CAB about the court bit. It may have changed but here’s my experience.
The first step was a letter sent recorded delivery explaining the situation, stating expected outcome, setting a deadline, and giving intention to go to court if not paid per the deadline. Make certain you have the confirmation of receipt.
If the deadline is missed then take it to court. I ended up with a hearing where the ex-employer made all sorts of crap excuses, a decision was made, then court adjourned. The adjournment gave the ex-employer fat git the opportunity to pay up without getting a judgement against him (there would be significant financial implications for future borrowing etc). The money arrived soon after.