My daughter graduated from a US college prep HS in 2016. She thougth for sure she would love studying in the UK) (having attended Latin camp in UK for three summers) and ultimately applied to St. Andrews, UCL, KCL, and Oxford. She was accepted everywhere but Oxford (rejected after interview). She decided to attend UCL and hated it. She and the only other incoming US student she knew prior to starting UCL each transfered to a (pretty high prestige) US university for sophmore year. Oh well, I loved visting her in London.
The general warnings cited here are directionally correct. It is difficult to say whether the specific issues experienced by my daughter (to the extent she is able to even articulate them) were her own issues, UCL specific, or a function of UK schools generally. My wife and I (both University of California grads) joke that UCL could easily have stood for Univeristy of California, London. For those who are familiar with the meat-grinder that is UC Berkeley, that is saying something. On the bright side, her tutor was very helpful with the transfer process and for that I am grateful. On the expense side, accomodations in London are very, very expensive. I paid $2000/month for a half of a tiny (500 sf) apartment next to TavistockSquare.
I will also warn you that American Latin honors kids (my daughter scored a 5 on her Latin AP exam and high 700’s on the Latin SAT subject test) will generally be far behind the best public school Latinists who have been studying Latin since age 6 and focused on Classics for two years preceding university during sixth form. Although she enjoyed not being at the top of a class, it was eye-opener for my daughter.
As for the specific schools, it is after October 15 so presumably you have already submitted your Cambridge application. Try to find someone to help prepare for the interview.
Of the other schols you mentioned, I have heard kids speak highly of Durham (both culturally and academically).
I have heard that US students love St. Andrews because it combines some of the advantages of UK uni that you have cited with a real effort to create more of an American college feel. The Classics course at St. Andrews is 4 years - but you will gradute with an MA.