The National Archives provides useful information and training for fundraising for archives and provides guidance on writing project bids:

http://www.nationalarchives.gov.uk/archives-sector/whats-happening-now/fundraising-for-archives/

The Wellcome Trust has been one of the most generous supporters to improving access to health related library and archive collections, chiefly through the Research Resources Grant scheme

https://wellcome.ac.uk/funding/research-resources-grants-support-libraries-and-archives

A list of recent projects that the scheme has funded can be found here:

https://wellcome.ac.uk/what-we-do/directories/research-resources-grants-projects-funded