It only took a couple of weeks last year, after the correct department was (finally) identified. Had to explain more than once that the form needed from HMRC was the proof of contribution to UK national insurance to satisfy the US requirements for documentation per the "US-UK Totalization Treaty", for self-employed Americans living in the UK. The certificate they sent is titled, "Certification that you're under United Kingdom (UK) Social Security laws self-employed in the UK", and it is identified as form "CA3940 HMRC 03 20" in the form footer. It provides:
1. US social security number
2. UK National Insurance Number
3. Your name and address and a daytime phone number
4. Date residence in UK began
5. Date self-employment in the UK began
6. Type of self-employed business
At the bottom of the form the following notification appears:
"We've sent a copy of this form to:
Office of Central Operations
Social Security Administration
PO Box 17741
Baltimore
Maryland [removed]
USA"
The cover sheet contains this contact information for that office:
HM Revenue & Customs
NIC&EO
BX9 1AN
United Kingdom
Phone +44 (0) 3000 555 724
Mon - Thursday 08.00 to 17.00
Friday 08.00 to 16.30
Fax +44 (0) 191 22 59570
hmrc.gov.uk
I'd advise calling the above number (telephone, not FAX, of course) and explain exactly what you need to them, using their form name described above. Good luck!
