Thames Riverboat to Hampton Court, Heathrow & Windsor

Yes the bus to Legoland runs from Windsor Parish Church 3x per hour- services 702 and 703.
If on the 703 ensure that it is destination Bracknell as in the other direction it goes to Heathrow Terminal 5.
So every half hour there is a bus from Legoland to Heathrow T5 until late at night.
The 702 goes to London Victoria and terminates at Legoland.

These are modern double deck buses.

What are you doing with your luggage while at Hampton Court?

There is also a red London bus, the #111 from Hampton Court to Heathrow every 10 minutes.

Sounds like a wonderful trip! My only caution would be to prepare yourself for a potential rainy day/bad weather on the boat day (particularly if you are going in early spring)- our past few springs have been particularly cold/rainy, but you can still enjoy the boat from the inside. You probably know this already but I would book all the entrances in advance so you don't have to queue.

Just be aware that the cruise from Westminster to Hampton Court takes between three to 3.5 hours. I found it - dare I say - rather dull and frustrating as it cut into my HCP sightseeing time. I’d rather spend longer at the palace. If I did it again I’d just take one of the shorter legs, say Kew to HCP.

With your flight leaving from Heathrow in the early afternoon on your day of departure, you could stay at a hotel in Windsor for two nights instead of at the airport. It's a very short (and inexpensive) taxi ride from Windsor to Heathrow.
This gives you time for breakfast at your Windsor hotel on your day of departure, then a taxi ride right to the door of the correct terminal.

You will need to decide how to manage your luggage while at Hampton Court Palace. There are lockers there, but not big enough for suitcases

LUGGAGE: We are travelling light, carry-on bags only so I was hoping there might be some sort of lockers.

WINDSOR v HEATHROW HOTEL: My first glance at hotels near Legoland suggested the best deal might be near Heathrow. Happy to stay in Windsor for the two nights if it's not much more expensive.

LUGGAGE: We are travelling light, carry-on bags only so I was hoping there might be some sort of lockers.

As I noted in my post, the lockers are not big enough for suitcases, unless they are really, really small. Most of the lockers are 45×45 cm (17.7 inches) there are a few that are 50cm (19.7 in).

Yup, I got that.

Carry on size limit is 55cm x 23cm x 40cm so we might be able to squeeze them in the larger ones. I have also started looking at luggage storage options nearby which might work.