Given the outrageously expensive tuition at top American universities, one would certainly hope that financial aid assistance is available.
Tufts is not an acronym; there’s no need to capitalize it. To answer your question, any of those can get you into graduate school, and it’s really a matter of deciding where you fit, as that’s where you’ll perform the best. In terms of pure academic heft, I’d rank them as LSE and UCL > Bristol = Tufts ≥ St Andrews ≥ Bath. This is something of an apples to oranges comparison, however, as Tufts is noticeably stronger at the undergraduate than graduate level, and comparing academic strength in the UK really depends on what you want to study. There are no bad options here.
What is your subject of interest? If it’s international relations, St Andrews has a very strong program, right up there with UCL and LSE and behind Oxbridge as the best in the UK. Tufts obviously has a very strong program as well.
There are some pretty big differences between your schools that have nothing to do with academic quality. Tufts has a lot fewer undergrads and offers smaller courses and more flexibility than the British universities. It also has more distribution requirements, such as arts, math, and foreign language requirements. Maybe this appeals to you; maybe it doesn’t. Don’t underestimate the impact of setting either; studying in London and commuting to LSE/UCL is a VERY different experience from living in a quieter city like Bath (which is lovely) or a small town like St Andrews.
The British universities have the advantage of being noticeably cheaper (assuming you’re a UK/EU citizen), both in tuition and because the degree often takes one less year (St Andrews is 4 years like the US). Moreover, UK university loans – unlike American ones – only become due once you’re making above a set salary, currently £21K.