We are arriving by rail to HOT on a Tuesday and plan to visit the rowing museum. Later that same day we are going to Heathrow Terminal 4. The train is expensive. Is there a bus from HOT to Heathrow?
Maybe not directly from Henley-on-Thames, but you can take a local bus to Reading and the airport bus from there. However, unless you have passes or discounts, the Reading Rail Air bus costs £17 each, and you would have to add the cost of the local bus to Reading. The train---with a transfer to the Heathrow Connect at Hayes & Harlington--- costs £17.20.
We did that trip by train last October after a couple of nights at Henley and it was fine. But maybe the locals will have some alternate ideas or suggestions.
Where are you starting from that day? If it is somewhere in or near London you may be able to buy an Off-peak Day Return to HOT for just a little more than a Single (typically only 10p more) and it will cover your journey back to Slough or Hayes & Harlington, leaving just a small extra fare from there to Heathrow.
My advice to anybody travelling from Heathrow Airport to Henley on Thames is NOT to use the route provided by National Rail Enquiries as it encourages you to use the Heathrow Express from Heathrow to Paddington. Instead, take the regular Tube (Picadilly Line) from Heathrow to Paddington via Hammersmith or Earls Court and get to Paddington that way. Then get the train to Henley on Thames. It will be much cheaper and may not be much longer.
In fact the same goes for any rail journey from Heathrow. On another occasion, National Rail Enquiries advised me to take the Heathrow Express to Padddington to get to Liverpool Street Station. Instead I took the tube (Picadilly Line) to Holborn and then the Central Line to Liverpool Street which was much cheaper and few minutes (not much!) faster.