I am not really sure what you are after. On the face of it, the question of "how much of the Peter-Weyl theorem survives for arbitrary locally compact groups?" admits the bald answer "Not very much" - there is a standard example of the free group on two generators admitting a unitary representation which admits a direct integral decomposition in two essentially different ways. (I first saw this on page 182 of Robert's book "Introduction to the representation theory of compact and locally compact groups""Introduction to the representation theory of compact and locally compact groups" but unfortunately don't know of other, more easily accessible sources.)
But perhaps you are more interested in algebraic groups, or Lie groups? Well, my very limited understanding is that people have been bashing their brains against this for over 60 years. Supposedly the case of semisimple Lie groups is the easiest one to get a handle on. (It is relevant that semisimple Lie groups are Type I, so that one doesn't get the same "pathology" of non-uniqueness of decomposition that afflicts the case of the free group.)
Perhaps this overview/advertoverview/advert may have pointers to some literature or background that answer your question better.
