When we were in London a couple of summers ago we called Great Western Paddington (probably on the advice of the hotel) for reservations for a day trip from London to Bath.

Did it maybe a day or two before we left for Bath. No problem at all getting reservations. Don't know if we paid too much...it was kind of spur-of-the-moment thing. Cost us a little less under $50 each. They take Amex.

Chris in Pearland, TX