Yes, there are many mutants who can heal as effectively as Wolverine, but most of them are directly related to him (either by blood or their powers being borrowed/copied from him).
As for mutants who can heal that fast that aren't in the Wolverine family, the only standout from the Marvel Wiki's 'Regeneration' category is Nathaniel Essex, better known as Mr. Sinister.
Mr. Sinister's healing ability tends to actually be faster than Wolverine's. He frequently takes a bullet (or ray blast or laser shot) and shows no more than a brief impact wound, which heals virtually immediately. He rarely bleeds from it (unlike Wolverine) and doesn't even seem to feel much pain.
He's not someone you usually think of as a regenerator because he is vulnerable to the Summers family of energy projectors. Cyclops and Havok can both cause him wounds that he doesn't immediately no-sell, and which do visibly cause him pain.
This has been consistently shown in the comics and even in the FOX Kids X-Men show from the 90s (which did tend to be fairly accurate to the comics):
If you expand the question to non-mutants, there's an even better answer:
Hulk is not just the strongest one there is, Hulk can heal almost any injury.
Hulk's regeneration is fast (faster than some versions of Wolverine, that's for sure!) and can regenerate him from some extremely severe injuries (as seen above).
He's healed from some pretty nasty things.
If I had to guess, I'd say his healing is like his strength: the madder he gets, the faster it goes. And pain tends to make him mad.
Given the frequent disparities in healing ability and speed seen in Wolverine and Hulk across different comic issues, comic titles, media, etc, I don't think a definitive answer can be made as to who heals faster (or who can heal better). That said, Hulk is at least a contender.
As others have said, so are X-23, Sabertooth, Deathstrike, and a number of other Wolverine/X-men supporting cast and villains. And Deadpool.