There are some recent papers of Valette and coauthors which give explicit calculations/descriptions of the K-theory for some of the Baumslag-Solitar groups, namely BS($1,n$) (arXiv 1604.05607), and also for certain lamplighters over ${\bf Z}$ (arXiv 1610.02798). In both cases, the authors determine the LHS and the RHS of the Baum-Connes "picture" separately, and then verify explicitly that the BC map is an isomorphism.

(Apologies if these examples are covered in the references already provided by Matthias Wendt.)

There are some recent papers of Valette and coauthors which give explicit calculations/descriptions of the K-theory for some of the Baumslag-Solitar groups, namely BS($1,n$) (Pooya and Valette, arXiv 1604.05607), and also for certain lamplighters over ${\bf Z}$ (Flores, Pooya and Valette, arXiv 1610.02798). In both cases, the authors determine the LHS and the RHS of the Baum-Connes "picture" separately, and then verify explicitly that the BC map is an isomorphism.

(Apologies if these examples are covered in the references already provided by Matthias Wendt.)