Private Tour Company

You don't need a tour company of any kind to visit Windsor Castle. Take the train, easy to do.

As Stratford upon Avon has a Shakespeare tour that visits all five of the historical Shakespeare sites that includes a tour, you just make you own way using the provided transport.

London Walks offers private tours to Windsor Castle. They mention that they can "arrange a private chauffeur."

https://www.walks.com/other-tours/private-walks/

Unless the elderly person in your party has severe mobility issues (other than just age) I don’t think you need a tour. The central station in Windsor is pretty much directly opposite the castle and a car couldn’t drop you much closer.

Thanks everyone. I know that we could do each of these as separate trips on different days from London, but we wanted to link them together and just make it easier for my father. I'll take your advice and look at trains. Thank you.

Oh my, I don't think doing both on the same day is a good idea!

For myself Windsor takes 3/4 a day. Haven't been to Stratford-upon-Avon in years so no idea how long that would take.

Does your Dad feel like he wants to see both? I'd probably choose Windsor as it's less of a fuss to get to.

It takes a FULL day to see Stratford upon Avon, doing the five sites with the Shakespeare tour and general covering the wonderful town.

By train it is effectively impossible to do both on the same day.
Even by tour bus or private driver you are pushing it.

Many thanks for the advice.