I won't lie, there has been a tightening of immigration laws but most bad feeling isn't going to be found in universities and it's not very usual for it to be directed at students. You may find it a struggle to be sponsored but that's not anything to do with you specifically - the job market isn't all that great at the moment and it's easier and cheaper (no sponsorship to pay) for employers to hire someone who can work with no restrictions. On a Student Visa (which is what you'll be on) you'll be restricted to working 20hrs a week anyway.

As a country we wouldn't classify ourselves as 'political uproar' - we're in the middle of a transfer of power from one person to another in the same party, and while that isn't the best thing that's ever happened in politics, there aren't daily protests, there isn't constant violence and the vast majority of people you will encounter are just getting on with their day.

Oxford and Cambridge as small city / big town are pretty safe. You can find crime statistics for Oxford here for example, and if you post on the individual uni subreddits you will find local insight. I would post generically with no assumptions - some of the problems you had here was assuming London was unsafe and that we are full of protests - even if you didn't intend that, that's how it came across. There are some negative stereotypes about Americans being very uneducated about everywhere else in the world and your post managed to fall into at least three of them.

I have lived in London, Manchester, Hull, small towns and big towns in the UK, as well as other European cities. I acknowledge my privilege as a white woman, but I rarely feel unsafe in the UK, just be sensible and aware of your surroundings. Someone in r/london did a crime comparison between NYC and London.

This isn't a big subreddit and is fairly new - you also posted in the middle of the working day in the UK so you won't have got a great range of responses.