I think a photocopy would suffice but they make it quite clear in the Mail on Sunday that
photocopies will not be accepted. I'm not sure about the numerous other publications but if they don't then they should make it very obvious.
I believe the Telegraph will accept a photocopy of the original grid, but not a photocopy of a completed crossword. So I suspect that other publications will follow a similar line.
Don't assume that ANY paper will accept a photocopy. Even if they state they do, if you were drawing the winner and a photocopy came out, wouldn't you pick another winner who'd actually bought the paper?