Firstly, I asked Hattie Hayridge, who played the ship's computer, Holly, from Series 3 to Series 7. Although she has not been in Red Dwarf since Series 7, she said that she would like to take part were there to be a crossover requiring Holly. She also revealed that Mark Gatiss is her neighbour, so she would hope he would get her a part in a Doctor Who crossover.
Chris Barrie, who plays Arnold Rimmer, though it was not very likely to happen. He explained that there is a 'bridge' between the Red Dwarf and the BBC, who often shunned shows like Red Dwarf saying sci-fi didn't work. But he said that were it put to him, he would happily join in. His final words on the matter were:
"Stranger things have happened... and all the other clichés."
"I think even just a brief thing where Kryten walks into the TARDIS and says 'Oh, wrong destination sirs'. I think that would be great."
Llewellyn also added that he has previously had lunch with former showrunner Russell T Davies and Doug [possibly meaning Red Dwarf writer Doug Naylor] so the crossover idea has been discussed before, but never amounting to anything.
In summary, it seems highly unlikely that we will see the Red Dwarf crew board the TARDIS any time soon, but if there were a Children In Need skit perhaps, they would be happy to take part.
Tomorrow we'll be bringing you an exclusive interview with Wendy Padbury.