Hi, I've moved your thread to the UCAS & Applications forum - although the Edinburgh course is an MA, it's formally an undergraduate course, not a postgraduate one (they issue the MA as the first undergraduate degree for reasons of tradition, in common with Oxford and Cambridge).
For a lot of your questions there isn't an objective answer and it really depends on what jobs you want to go on and where. They will also depend somewhat on your own personal preferences - some will argue that social life in London may be worse than other university cities as although there is a lot more to do, students tend to be more fragmented across the city, everyone has less money to spend on social activities due to how expensive it is, and the universities there (including UCL) are very large and can alienate students who would prefer a more intimate social scene (although this latter point probably also applies to Edinburgh).
Generally if you are aiming at working in the UK after graduation, for pretty much any field except investment banking or management consulting university "brand name" or "reputation" counts for very, very little (if anything). For those two exceptions though it's pretty important to go to a banking "target" uni - which UCL is and I believe Edinburgh is not (I think it's considered semi-target?). For further academic study the two are pretty broadly comparable (one may be better than the other in a particular niche area vs another, but this will depend very much on the exact areas of research/study under consideration).
I'd note the two degrees are rather different in themselves as well. The UCL one specifically focuses on North and South America in historical and political perspectives, using both to build up a picture of the Americas as a whole across modernity broadly. At Edinburgh it's a typical joint honours degree in history and politics, so the two sides are more or less independent, and you have flexibility to study disparate areas in each if you like - you could well combine some medieval Scottish history with modern Japanese history there for example if you wanted to, across your degree, with various politics options on the other side. Whereas at UCL the modules for the degree are primarily run through the Institute of the Americas on both the politics and history sides, so there is more of a thematic link running through them all.