Well, if they ever did an animated version of Azkaban, most certainly my foremost choice for Sirius' voice would be Alan Doyle of Great Big Sea. Sure, he's not British, but he does have that good ol' Irish accent of St Johns. He wouldn't work for a live action film, though. I've said it before and I'll doubtless say it again: he's too soft-looking. The guy couldn't look murderous if his life depended on it.
As for Remus: Darrell Power, both for live action and animated. He'd make the perfect rumpled-tweed Professor! Though perhaps his voice is a bit too light.