The Big Red Bus tours are another idea for your day. If I recall correctly, you can buy a ticket with on-off privileges good for 24 hours. On the first day, when you are tired and jet lagged, you can just ride around the circle, seeing everything and making notes of what you want to revisit. You can use the bus almost as public transportation to get you from place to place. There are also bikes you can rent, a ride through Hyde Park in August would be lovely! And although you said you're not interested in shopping, it's worth wandering past/through Harrod's just as a sight-seeing option.