I sent my NI38 and cover letter off on 21st Feb 2022, on 6th May they had calculated my "buy-back". I paid on 27th May 2022 and followed up by phone on 8th September 2022 to see about the crediting to my record, that's when the agent said he would send a note to HMRC for international caseworker, this normally would be "urgent". I received a letter this week, dated 4th October to confirm the allocation. So I feel they did complete the last part reasonably quickly, but my call on the 8th September was important.