Ah, an MSc...it's the "equivalent" of an MPH here. I haven't heard of Liverpool, but
London School of Hygiene & Tropical Medicine is known worldwide for their research and prestige. Why not apply there? In terms of name recognition, it depends on what you want to do after you graduate. If you want to work outside of the UK, I would just make sure to find an accredited program (for the US it's CEPH) if applicable. You're paying money for a program, so you want to make sure it's recognized as one that meets certain standards by potential employers. Something else you can do to evaluate the program is look at their alumni page if they have one and see if they name the workplaces or countries alumni go after graduation.