I think officially (i.e. based on what the Rail Delivery Group -- who are, i I believe, the ultimate authority on this -- have told me), they want to see a passport. The T&Cs also do state that a passport should be presented for the initial validation at a ticket office.Should I be worried that this won't be accepted?
In practice, I think the response might vary and I expect many ticket officials would go with the spirit not the letter of the law. But I think you might find one or two (the consensus seems to be -- beware of manual ticket checks at London Euston!) who insist on the passport as that is what they have been told must be shown.
I think your hardest case to argue (should an official cause difficulty) is the initial validation check, as the terms and conditions do state a passport must be taken to a ticket office. I think it would be easier to argue that there is no specification that a passport be shown at any subsequent ticket check.
It is probably worth calling up the company that sold you the ticket to ask this question and get an answer in writing. Does anything in their sales material specify you need a passport to buy the ticket, or just foreign residency? If just residency (which seems to be what most vendors state), it seems a very legitimate question to ask them how to proceed without a passport.