I would say no. Even with a layover and not having to get your luggage, you have to go through customs. That will take awhile. If you are doing the tour by yourself, you can ask if you can do it at 9:30 or ten. They might be able to accommodate you. Also, our flight was delayed three hours getting into London, so the plans all fell apart. So maybe have a Plan B.

We booked a tour on the afternoon of our arrival. It worked well - we had enough time to get to our AirBNB, drop off our luggage and splash water on our faces, and the car picked us up at the house. Look at Google Maps for transit info to get from London to Southampton. I've not traveled that way, but it looks very do-able. You can also rent a car (which is what I've done). We enjoyed the freedom!

Yes of course you can get a four hour London tour. Me and my family we had a fantastic time. Four hour one to one with tour guide it is quality tour

The change of guard is not everyday and not at the same time, you should check the schedule on line. If you are planning to take the private 1/2 day tour, I would take it in the morning, 9 am should be a good time.

I just got a tour from them within the past two weeks. Don't know about the toddler. We paid for our 10 year old, but that is obviously different. They are quite accommodating. Remember that the tour is only about a half day, including driving. So in depth going over individual attractions is unlikely. I would ask the driver to include the particular attractions, and as long as they are not far from central London, they will do it. At least that was our experience. They will also try to cater to your interest.

Mine was four hours but I understand that there is some flexibility for a shorter or longer tour