Late Payment and no reply. How to Proceed?

Good afternoon,

I write here asking for advice regarding a late payment.

I am a freelance designer, and I had already worked with this same customer previously. The client Accepted my cost estimate, then accepted (Writing) the work when delivered. I invoiced him for a total of £3240 on the 1st of March, payment due in 7 days.
Since then I have not receive any reply, nor the payment. After numerous email, texts, unanswered calls and voicemails, on May 2nd I sent him the 1st letter by post, 1st class signed for, with a bold red "URGENT PAYMENT REQUIRED" on top, containing a letter where I said that the amount was still overdue and that i was now asking for minimum late payment fee of £70+ interests, payment in 7 days or I would proceed to the court. A copy of the invoice and the new statement were attached.
After 8 more days of silence, I sent another "sign for" letter, with the same updated info, invoice and statement.
What could I do now? How do I proceed now?

Thank you

Hi James,

thank you so much for your prompt answer.

Would you suggest to do it myself or to ask for a debt collection agency? I fear the fees of the agency, but maybe somebody can reccomend a trusted one? Also, what will happen if this guy is not reacheable?

thank you