Yorkshire MESMAC traditionally has focused on fighting HIV in the Men-who-have-sex-with-men community, but they offer testing services to everyone and their outreach is now more broadly focused on anyone who's LGBTQ. They offer weekly testing for free without appointment. Check their website for the most up to date times.
Access seems to be primarily through a door from the street with a very high step, followed by a flight of stairs and operates on 2 floors. There wasn't any immediately visible alternative access option, like lifts or ramps.