The express train in from Heathrow to Paddington is £21.50 single or £35 return and is not included in any passes whatsoever. You can use the London Underground service instead which is cheaper but slower.

I don't normally encourage the "London Pass" and similar products unless you're going to be going to multiple attractions most days. Online offers tend to be very competitive when you book directly with the attraction, and then you don't need to wait in line.