I would also like to thank everyone for their contributions, especially the OP.
A relative of mine, approaching retirement, had lived in the UK several years as a child and worked there for less than a year as an adult. They didn't know their NI number, so I brought them through the whole process, mostly out of curiosity. Neither of us expected anything to come of it.
We found four years of credited contributions on record, and submitted the CF83 form online in mid-August of this year. By the end of September they got a letter of eligibility to buy back to 2006 at the Class 2 rate. That, plus the next couple of years, should bring them to at least two thirds of the annual UK pension by the time they hit retirement age, for less than £4k total cost.