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: (i) in Mathematics and in Theoretical 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.

Marie Curie also supports researchers at levels other than postdoctoral. (This second link looks more up to date)

Royal Academy of Engineering have Postdoctoral Research Fellowships with an application deadline of October.

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.

Cambridge College Research Fellowships

Using their endowments, colleges at Cambridge offer a number of Post-doctoral Research Fellowships aimed at people close to completing their PhD or recently having completed it. In total, some 200 research fellowships are available each year across all subjects. Most of these are stipendiary and offer a salary for three years and rooms and board in college. Normally, the only requirement is that the research fellow spends his/her full time on research. Competition is intense and successful candidates are usually ranked as 'outstanding' by all referees. Contrary to widespread belief, they are mostly open to graduates of any university and there is no preference given to Cambridge applications. Post-docs (or PhD's close to completion) wishing to join a Cambridge Group and apply for college research fellowships should let the group know their intention but should apply using their own referees. Details of research fellowships are (only) published in the Cambridge University Reporter. This is published weekly and is available on the internet (look for the annual Awards Special Edition). It is worth checking this throughout the year for research fellowships are advertised at different times (though mostly between August and December). Colleges are generally inundated with applications and you will find that they expect you to check the advertisements and are unlikely to agree to send you details when they become available.