If you haven't already bought your tickets, note that sometimes it can be cheaper to buy one through ticket for the whole journey, whereas sometimes it can be cheaper to split tickets. I suggest using the raileasy/split-ticket web-site:
https://raileasy.trainsplit.com/default.aspx
If you buy a through ticket, make sure that you specify via London if you want your ticket to be valid that way. You may need to select "advanced options" or such like on a web-site. If you buy a through ticket via London, then your ticket should include the tube fare, as explained by jBuchanan.
If you split tickets, and buy separate Bristol to London and London to York tickets, then rather than buying a Bristol to Paddington ticket, you could buy a Bristol to London Zone 1 ticket which would include the underground. Or alternatively you could buy a London zone 1 to York ticket. The Raileasy site will do this for you.
Note that tickets that include the underground should only be issued on the traditional orange-card tickets, for collection at a station or posted to you. Do not buy an electronic ticket or print-at-home ticket if you are offered these options, as these tickets won't work the underground ticket barriers, and are unlikely to be accepted by the underground barrier staff.