One and 1/2 day in London suggestions

Have you been to London before?

If your train is on time, it will probably be 3.30 by the time you have checked into your hotel and you are ready to start exploring.

Organised walks are a good way of learning about a place and there are many, catering to different focus groups eg general interest, food, pubs etc.

Oyster cards are valid on the Gatwick Express, but not the HoHo buses. I would prefer to take a seat upstairs on a regular London bus such as the number 23 that runs between the City and the West End rather than take the HoHo.

What are your interests for seeing things on the Thursday? Cathedrals/ shops/ WW2/ Parliament/ royalty/ museums?

You will have to be at Gatwick by 7am latest on Friday morning, which will mean an early start, away from your hotel by 5.30am unless you stay at Gatwick the night before.

The Oyster card, loaded up with "pay-as-you-go" money, is valid on any public transport within London. Tube, rail and buses. And, as an exception also the rail link to Gatwick.
It is not valid on Hop-on hop-off buses or bus tours, these are run by private companies for tourists, not to get Londoners from A to B and are a lot more expensive than normal buses. I am not sure what you mean by "the express", please clarify.
"I'm told the bus stop" - which bus stop? There are many rival tour bus companies.

A hop-on hop-off or other bus tour with commentary is a good idea when you first arrive in a city like London. It lets you get an overview of the geography and where the main sites are. After that it is a lot more efficient (and cheaper) to get around by public transport (buses, tube and train).

For understanding London's Public Transport read the following:
1) Info leaflet for tourists: http://content.tfl.gov.uk/london-visitor-guide.pdf
2) The famous tube map: http://content.tfl.gov.uk/standard-tube-map.pdf When looking for hotels, the important thing to know is what the name of the nearest tube station is, and how far away it is.
3) Key bus routes in central London: http://content.tfl.gov.uk/bus-route-maps/key-bus-routes-in-central-london.pdf his is not all the bus routes, just some that connect tourist places in central London.

Where is your hotel? Do you arrive at St. Pancras? What are your interests?

I have a suggested itinerary for Wednesday if you want to hop right in. If your hotel is close enough that you could get there and check in by 3:00 and head straight out the door, you could take the tube to St. Paul's. If you could arrive by 3:30, you could tour the cathedral and stay for the lovely evensong service held at 5:00. After that, stroll down the strand to Trafalgar Square. Have dinner on the way or eat at one of the many pubs near Trafalgar Square. From there, walk down to Westminster and see the Houses of Parliament, #10 Downing Street and the river. Have a drink if you wish, then catch the tube back to your hotel.

Thursday will be a series of hard choices. The weather may be a deciding factor on whether you do outside or inside (museum) activities. The problem is an activity like visiting the Tower of London will take up half a day.

A lot has to do with how you like to see the sights. If you like tours, I'd check to find a "London in a Day" tour for Thursday. My wife and I prefer to go it alone. You could do Hop on Hop off or even just get an Oyster card and combine the bus and tube. Are there just two of you or a bigger group? I would imagine a tour would be easier for more than two from a logistics standpoint.

We plan to be back in London in June as well. My wife is being inducted into the (Florence) Nightingale Society. It should be quite a ceremony since it will be held in Westminster Abbey!