One reason might be that by travelling in the TARDIS, Wilf picked up some 'time-traveller-iness' which enabled him to remember things that others would forget. From Series 5 Episode 5, Flesh and Stone (emphasis mine):
DOCTOR: The Angels all fell into the Time Field. The Angel in your memory never existed. It can't harm you now.
AMY: Then why do I remember it at all? Those guys on the ship didn't remember each other.
DOCTOR: You're a time traveller now. Amy. It changes the way you see the universe, forever. Good, isn't it?
Wilf travelled in the TARDIS, like Amy, during the episode The End of Time, Part 1, so he might have gained the same attribute of not forgetting things which others forget.
Two counterarguments to this theory, and my counter-counterarguments:
Wilf never travelled through time. The Doctor told Amy that she was unable to forget because she was a time traveller, while Wilf never was.
But maybe travelling in the TARDIS is enough. As far as I know, the TARDIS draws its energy directly from the Time Vortex, so even travelling through space in it might be enough to pick up some traces of Time Vortex energy and lose one's ability to forget certain things.
Wilf's trip in the TARDIS came after his dreams. The passage you quote in your question comes from near the beginning of The End of TIme, Part 1, while Wilf only travels with the Doctor much later in the episode, after he's already had his dreams and remembered them while everyone else forgot.
But time travel is a tricky thing. For all we know, the memory abilities granted to time travellers may extend throughout their timeline, not just into their future. After all, the terms "past" and "future" are less meaningful when time travel is involved. We've seen this effect before: for example, the creation of Jenny in The Doctor's Daughter retroactively caused the TARDIS to travel to that place and time so that she could be created.