My info is a few years out of date, but it used to be possible for private individuals to get film processed and printed at various University departments around Cambridge. The Engineering department was one, and there used to be a central photo lab on the Downing site. A quick search of the university website turned up this lab:
Otherwise, the members of the Cambridge Camera Club (cambcc.org.uk) can probably give up to date info. Campkins also used to employ generally helpful and knowledgeable staff.
I used to use Colour Quick in Regent's Street. Despite the name, and the cheap and cheerfull retail style, they had a very good traditional wet lab, and a custom printer who worked hard to get things right for reasonably cheap handmade enlargements. Again, I last used them a few years ago now, but it was for my wedding photos, and they did a great job.
That said, I have used Peak Imaging for many years now and they too do a good job. B+W prints are via digital printers onto colour paper though.