Some Funding Sources for Postdoctoral Research in the UK

Royal Society University Research Fellowships: 5 years (plus possible extensions to 10) for EU citizens who will have 2-7 years' postdoctoral experience at appointment, are not already in a substantive post, and are either currently employed in the UK or have resided in the UK for at least 3 continuous years not for full-time education. Also on this page, Dorothy Hodgkin Fellowships which are for 4 years for EU citizens with PhD and at most 4 years' postdoctoral experience and the same other conditions.

Royal Society International Exchanges: Schemes listed by country (click "Country Index") for visits to the UK (of or UK scientists to your country).

EPSRC Fellowships (click on Research Support, then Funding Opportunities then Fellowship Opportunities): (i) in Mathematics and in Theoretial Physics: 2 years for EU citizens who have/will have PhD and at most 3 years' postdoctoral experience and are not already in a permanent academic post. (ii) Advanced Fellowships, which are for 3 years and are not restricted to EU citizens.
Also EPSRC accept applications for research grants from researchers at UK universities which can include requests for funding for a postdoctoral research assistant, who can be named in the proposal, but my current personal perspective on these is that the work for me to write the proposal, get the postdoc to do the proposed research rather than what they (quite reasonably) decide they are more interested in at the time, and write the final report, usually outweighs the benefits to me. EPSRC also accept applications from UK researchers to host Visiting Fellows and these are less trouble.

EC Marie Curie Individual Fellowships: 2 years for EU citizens (or 5 years residence) not a national of host country nor worked there for more than a certain time.

Leverhulme fellowships and exchanges

British Council: various opportunities depending on your country

London Mathematical Society: small grants for visits (select "Grants")

Royal Commission of 1851 Fellowships: 2 years for citizens of UK, Commonwealth, Ireland or Pakistan, about to or recently obtained a PhD.