There are almost no meaningful scholarships for international undergraduates in the UK (they likely won’t understand any terms like “merit aid” so it’s a waste of time using them). The business model is that foreign students pay higher fees to subsidise the home ones.
There are lots of foreign students who have funding, but it’s always from their home countries (eg Singapore government). You probably won’t get anything from the UK.
The only thing you can get at Cambridge is a Cambridge Trust scholarship.
https://www.cambridgetrust.org/scholarships/key-points-for-undergraduate-applicants/
The main issue with this is proving financial need. That is pretty difficult for an American. You’re probably really wealthy compared to all the applicants from India, China etc. The other problem is you have to apply an be accepted at Cambridge before you can apply for this award. So you have to apply in October, receive an offer in December/January, apply for a scholarship, receive a scholarship offer in April and then meet the offer conditions if there are any. It’s a long process.
As noted above, some colleges do provide financial awards to students, but these are very minimal. The key words are “partial financial support”. This usually means a £200 book grant. Not a meaningful dent in £20,000 a year. There are also academic prizes. I got one of these from my college every year. It was £100, which is nice but won’t get you a student visa (you have to prove you have funds to pay for the entire 3 or 4 years of your course). I also once won a Biology prize - it was a subscription to “Nature” magazine!
Oxford provides some information about applying for loans.
http://www.ox.ac.uk/students/fees-funding/us-canadianloans
They also provide some links to external scholarships.
http://www.ox.ac.uk/students/fees-funding/ug-funding/external-scholarships
There is a lot more funding for graduate degrees (but quite possibly a lot more competition).
I presume Alcibiade’s D receives reduced EU tuition and is eligible for a UK government student loan on top of that. I know of no “bursaries”.
Finally, unless your son’s ECs or volunteering are directly related to classics, they are irrelevant to his application. He needs to show interest and focus on Classics. They don’t care about cheerleading club and the stray dogs centre etc. Unless he is an organ scholar, your son can only apply to either Oxford or Cambridge. You cannot apply to both for undergraduate in the same year.
Google is your friend by the way. That’s how I found those pages linked above.