I've been to Russell and non-Russell Group unis and none of them gave a monkeys if I'd volunteered or not.
However, if you are applying for a course that, perhaps, asks for previous experience (maybe one of the healthcare ones?) then well targeted volunteer work might be handy to have for any Uni. As DataVenia suggests.
Some employers do like to see volunteering activities on a CV, and I've been a STEM Ambassador for about 6-7 years now myself.