malmontar, my son has a fairly long journey to grammar school (but not in London). Last year (year 7) we went for the one bus option, thinking that it would be safer and easier for him - it was a school bus to another school, so he had a 10 minute walk at the end. He left the house at 7 and arrived back home at 5, so it was a very long day.
He managed ok overall. He did get tired, getting out in the morning over those snowy months was a struggle, but you'll be fine with the later start! The school bus was very noisy and overwhelming for him though, not particularly comfortable and being a bus in city traffic it was sometimes very late - I remember one snowy day when it took him about 2.5 hours to get back!
This year, he's doing public transport - lift to train station (5 mins), train, then tram, then 10 minute walk. He's loving it! He loves that there's lots of options should he miss the first train/tram (he often nearly missed the school bus!) and he feels grown up and independent navigating the links. He's also popped into the city centre for Krispy Kreme donuts in between his connections on the way home! In hindsight, I wish we'd switched to public transport halfway through year 7.
My general point though is that trains and trams are much more pleasant than buses and they don't get stuck in traffic so it's much less frustrating. I'd choose a train/tram journey over bus any day, so I think your comprehensive journey could actually be more tiring and frustrating.
I also have a daughter at an independent school and understand your worries about that! No school is perfect though, so you have to go with best fit.