I would pull up the curricula for the MSc and MPH (Harvard, Hopkins, whereever) and compare the core requirements.
I wish I could help you with this, but from what I can tell (from when I was researching schools in England a few years back) LSHTM is very highly respected, but not very well known in the States. I think the same could probably be said for Hopkins over in the UK/Europe. So, if you plan on working in the US after completing your degree, you may want to consider going to school here. If you plan to work in the UK or other parts of Europe, probably LSHTM or Harvard would be your better bets. They're all great schools, so I would look more at the curriculum and faculty as the deciding factor - Does the program allow for plenty of electives? Are there more than a couple profs who share your interests?