To complement Neil's answers: At Oxford and Cambridge you will find things called "Junior Research Fellowships". These are funded by and administered by the colleges (and not the mathematics department). They could be in any subject, or in, say, "Science and Mathematics", or even just "Mathematics" (but it's very unlikely to be more specific). Probably the college will rotate which subject(s) it wants over the years.
These position receive a very large number applications, and so some sort of filtering system along the lines Ben suggests must happen. Having strong references is perhaps the most important thing.
(I had one of these. I didn't know anyone at the college before, but in the end, a mathematics fellow at the college in question worked in a field rather close to mine, and interviewed me, so presumably understood my work and was sufficiently impressed by it. But the other postdoc in my year at the same college was in a different area, and the college simply invited in an expert from outside to interview him. So my strong advice would be to apply for everything, and get lucky once!)
The best guide to these things remains, I think, Tom Korner's: http://www.dpmms.cam.ac.uk/~twk/fellow.pdf
I think a few other places in the UK (Warwick? Imperial College London?) might offer one postdoc position a year under a similar format (any subject, best candidate wins)??