I would get some pounds from your bank in the US before you leave. The convenience would be well worth the small fee incurred.
If you go by bus from Heathrow to Windsor, you will approach the window at the Heathrow bus station, buy a round trip ticket to Windsor, and the clerk at the window will take a credit card for this. We used a Master Card. Ask at the window what bus number you will be taking. There will be lots of buses pulling in and leaving from the bus station.
You will wait outside the sliding glass doors at the Heathrow bus station for your bus. There are benches there. Watch for your bus to pull into the stalls. The bus driver will get out of the bus to help passengers with luggage, but will not yell out "Windsor" to announce this for people waiting to board. You will have to be alert and spot your bus number when it pulls in. Immediately board your bus; they don't wait very long.
If you take a taxi into Windsor, you will probably need cash, although some taxis do take credit cards.
ATMs at banks we have used in Windsor include Barclays Bank, at 29-30 High Street and Lloyds Bank at 2 Thames Street, both just steps away from Windsor Castle.
For admission to Windsor Castle, they take credit cards. For all the cafes and shops we went to in Windsor, they took credit cards. Costa Coffee, a small coffee shop, even took credit cards. When paying at shops and restaurants, be ready for them to ask, "Would you like me to convert this charge to U.S. dollars for you?" Always say "No." thereby avoiding a hefty conversion charge.