There's no need to use a specific card at the self-service machines. See this question. In fact, although the confirmation email says "please take the card you used to make the purchase", the recent machines I have used (at least Virgin West Coast and Arriva Trains Wales) merely say "please insert a card for confirmation purposes" - they don't specify it has to be the card you used to book with. You will however need the confirmation code, so keep that with you.
If you don't have a credit or debit card at all, then you have a few options. One is to try loyalty cards and other magnetic stripe/chip cards you have (if any). As the above question shows, at least Nectar cards have been known to work in some situations. It's possible that ATM cards and the like might work as well. The ticket offices are able to print out tickets purchased online, given the confirmation code, so that should be an option -they have asked me for a card for confirmation before, but they must have ways around it - and you are at least dealing with a real person now. Brining the original email and some photo ID can only help here. The ticket offices can also be useful when the machines have longer queues, or are not working properly. You could also ask for another person in the queue for the machines to use their card. People will probably be suspicous of some kind of scam (especially in London), but you can explain your situation clearly and let them do all the operation of the machine. There's a slight risk that they will run off with your tickets, but this seems pretty unlikley.
As a side note - unless you gain some specific advantage from thetrainline.com, you should be aware that they charge an additional fee to book tickets, which are identical to the tickets that can be purchased direct from the train companies - and all the train companies can sell you a ticket for any other - ie. there's nothing stopping you from having bought your ticket for a Virgin train from First Great Western. See this fantastic answer for a guide to UK tickets and purchasing them.