I hope I can point you in a few possible directions and also warn you off some of your ideas.
i assume from the late arrival in London that you are flying from the west coast, if so, the last thing you want to do is go to the Carnival, it is absolute mayhem, slightly less perhaps than on Monday when up to a million people will be on the streets, but nevertheless not something to consider when you are exhausted from travelling.
Hampton Court is an hour's train ride out of London, whenever there is a special event, and not only is there the Good Food show but Monday is a public holiday, it will be absolutely packed and very tiring. I would leave that to later in the week. London itself will be busy as it is the last week of the school holidays and many people travel from other parts of the UK for the long weekend. Seeing Big Ben is a 5 minute photo opportunity but it is on the other side of the road from Westminster Abbey so those fit well together, you could also take a river cruise from Westminster Pier or Charing Cross pier which is a 15 minute walk along Victoria Embankment. You mention the Thames Clipper, although they have by far the nicest vessels on the Thames they are a commuter service and do not offer a guided tour to tell you what you are seeing as you sail along, if you want a 'tour' in the real sense you need to use someone like City Cruises who will offer a commentary.
I do not know where you are staying whilst you are in London but 'platform 9 and 3/4 can be 'found' at Kings Cross station on the wall to the left of the ticket office. Having never read any Harry Potter I don't understand the attraction of seeing it and being photographed beside it. There is a baggage trolley seemingly coming through the wall and at peak hours someone hands out a scarf for particpants to wear and a professional photographer will offer to take a photo, no doubt at a very high price but at other times, you can just stand by the wall and take your own photographs.
Midweek I wouldn't waste time going to Borough Market for the food ( most stalls open Thur/Friday and Saturday) Camden Lock market is quite empty mid week and frankly most of what they sell is complete rubbish bought I am sorry to say by tourists, they can get 100,000 at the weekend, but mid week is even more of a waste of time.
Buckingham Palace is certainly worth seeing and you must book the tour in advance, most days are sold out (it is our Alcatraz as regards booking ahead) It is only open for 2 months a year, if you must see the Changing of the Guard, book a tour for 1230 or 1300 as the Changing at Buckingham Palace begins at 1130 and for most people all you see are other tourists heads in front of you as everyone strains on their toes to see what is happening.....
Piccadilly Circus and Trafalgar Square are only a few minutes apart, PC has a few shops and electric advertisements but the rest is just a traffic island, likewise Trafalgar Square is a large traffic island but with some great places ot visit around it such as the Natioanl Portrait Gallery. If you want to see a better Changing of the Guard you might walk down Whitehall from Trafalgar Square and see Horsegaurds Parade, you can get photos standing next to the guards on their horses very easily, a 6 minute walk from Trafalgar Square or Big Ben at the other end of Whitehall.
For you big day out you need to be aware of distances, Bath which is stunning and highly recommended is 114 miles from London, if you try to do too much you will have very little time there. I will admit I have never seen the attraction of Stonehenge, you cannot get close to the stones and yes, they may have been there millions of years but these are not the pyramids or even Mount Rushmore, they are random stones in a circle but if you must see them, you must. I also love Windsor, we are lucky enough to be able to visit any weekend we want and the castle is fascinating but you are planning a visit to the Tower of London, possibly the best use of three hours In London that I can think of, so do you need need to see inside another castle? If the answer is 'yes' then the only way to do both in a day is by a tour bus but be aware of just how little time you may have at each point. If you just want to get to Bath, there are trains every 30 minutes from Paddington Station which take just under 1h 30 minutes and if you book in advance you can buy tickets from as little as £14.50 pp each way, the later you leave it the higher the cost so even if you book it a day before it will be less than £56 which sesm to be be the cheapest return fare.
The station in Bath is close to the centre and there is a hop on hop off bus from outside the station to take you around the sites if you wish, all I can say is my Mom who passed at the end of last year enjoyed Bath more than anywhere else I took her in her 80's and would have loved to have spent longer than we could in the city. I wanted to take her to the Pump Rooms for afternoon tea where there is a small (very) orchestra playing while you nibble on scones and jam and sip tea but as someone who never had much money, she refused to allow me to treat her but I do recommend it.
If you want to fit all 3 into a tour then EvansEvans, Goldentours or Premium tours normally come highly recommended but it will be a 12 hour day on a bus.
When you arrive at Heathrow buy Oyster cards, they can be recharged as necessary but they guarantee you pay the lowest price for buses, the Tube and even a discount from Thames Clippers. you will never pay more than a maximum daily rate even if you use it 10 times a day, and you won't.
If you need any more help, please ask!