Below is an overview of Postdoctoral Research Fellowships in the United Kingdom, which was originally curated by Professor Rogier Kievit. These have a special focus on psychology, neuroscience, development and mental health. They vary in remit, duration and eligibility.
The colleges at Cambridge advertise junior research fellowships (JRFs) each year. Look out for these on college websites and in the Cambridge Reporter. Some JRFs are stipendiary (ie they come with a salary and sometimes modest research expenses, others are non-stipendiary (ie they do not offer a salary but may offer some benefits such as dining rights).
The Borysiewicz Biomedical Sciences Fellowship Programme
Parke Davis Exchange Fellowship [USA-Cambridge specific]
Daphne Jackson Fellowship [requires career break]
Leverhulme Early Career Fellowship
https://www.leverhulme.ac.uk/early-career-fellowships
Branco-Weiss Fellowship
https://brancoweissfellowship.org/
Jacobs Foundation Fellowship
https://jacobsfoundation.org/en/activity/jacobs-foundation-research-fellowship-program/
Human Frontiers Fellowship
http://www.hfsp.org/funding/postdoctoral-fellowships
MQ Fellows Award
https://www.mqmentalhealth.org/research-programmes/fellows-award/
Wellcome Early Career Awards
https://wellcome.org/grant-funding/schemes/early-career-awards
British Academy Postdoc Fellowships
https://www.thebritishacademy.ac.uk/funding/postdoctoral-fellowships/
UKRI Future Leaders Fellowship
https://www.ukri.org/what-we-offer/developing-people-and-skills/future-leaders-fellowships/
MRC Career Development award
BBSRC Discovery Fellowships
https://www.ukri.org/opportunity/bbsrc-discovery-fellowships-2023/
ESRC Postdoctoral Fellowships
https://www.ukri.org/opportunity/esrc-postdoctoral-fellowships/
ESRC New Investigator Grants
https://www.ukri.org/opportunity/esrc-new-investigator-grant/
NIHR Advanced Fellowship
https://www.nihr.ac.uk/explore-nihr/academy-programmes/fellowship-programme.htm
Royal Society University Research Fellowship
https://royalsociety.org/grants-schemes-awards/grants/university-research/
Royal Society Dorothy Hodgkin Fellowship [for ECRs who have caring responsibilities and/or require flexible working]
https://royalsociety.org/grants-schemes-awards/grants/dorothy-hodgkin-fellowship/
Royal Society Wolfson Fellowships
https://royalsociety.org/grants/royal-society-wolfson-fellowship/
Newton International Fellowships
https://royalsociety.org/grants-schemes-awards/grants/newton-international/
Royal Commission for the Exhibition of 1851 Research Fellowships in Science or Engineering
https://royalcommission1851.org/fellowships/research-fellowships
Brain Postdoctoral Fellowships
https://guarantorsofbrain.org/grants/post-doctoral-brain-entry-fellowships/
AXA Fellowship
https://axa-research.org/fund-your-research/post-doctoral-fellowships
Marie Curie
https://marie-sklodowska-curie-actions.ec.europa.eu/actions/postdoctoral-fellowships
NARSAD Young Investigator Grants [US-based but can be used in the UK]
https://www.bbrfoundation.org/grants-prizes/narsad-independent-investigator-grants