I don't have much to add, but as we also were classical homeschoolers I was intrigued & wanted to say hi ;-)
Both of our sons seriously considered uni in the UK, but ended up at (excellent) big state schools here (go figure!).
We spent a week in Oxford last month learning to ring tower bells (can't get much more English than that!) and had a fabulous experience. I agree that the Lakeside Guest House is in a good location (we stayed in a nearby flat); since it was dark and rainy much of the time we were there, we took the very convenient buses that run up that north-south road (it has various names) into town. We were also in Oxford in September and stayed at Christ Church College – highly recommended, of course!
It's definitely a great idea to visit in person. We met a young American woman (a new bellringer) doing her master's degree at Oxford; she had chosen an Oxford college at random when she applied, and ended up far from the city center. She was kicking herself for not having done more research when she applied. (She LOVED everything else about Oxford, including her college, but her dorm was quite a hike from the city center.)
Adding: in September we drove to Oxford (b/c we needed a rental car the following week for my husband's business trip) and that was a big mistake. In December we took a coach (bus) from Heathrow and it was perfect – as a PP says, they have WiFi and comfy seats – also a toilet, and the coach driver deals with your luggage. If you are coming from London a train also makes sense.