It's been a while since I read POA, but I believe he broke out in his animagus form, as the dementors were only used to guarding humans, and Sirius Black was unregistered so no one knew he could transform.
He escaped in his Animagus dog form. He didn't need a wand to turn into a dog - presumably a wand is only necessary to make yourself an Animagus in the first place. He'd been shifting into dog form repeatedly because the Dementors have less of an effect on animals, and in that form he'd become skinny enough to slip through the cell's bars. After he learned that Wormtail was at Hogwarts, he escaped and swam to shore.
