If you are happy to work with the Operator Algebra approach to quantum groups, then Woronowicz's definition of a Compact Quantum Group admits a perfect analogue of the Peter-Weyl theorem (looking at unitary corepresentations). The Wikipedia page doesn't give any references, but a good (and freely available) survey is: arXiv:math/9803122v1 [math.FA].
But maybe this isn't what you're after...
