We continue our investigations of the generalised braid groups appearing in 2d gauge theory, as fundamental groups of spaces of admissible deformation parameters ("times") for the irregular isomonodromy connections. Here we study the local wild mapping class groups in the twisted setting for arbitrary formal structure in type A. General configuration spaces will be defined and shown to admit product decompositions, via a suitable construction of fission trees. Moreover the fission trees will be shown to parameterise admissible deformation classes and used to visualise the configuration spaces. Simple examples give the braid groups of the complex reflection groups known as the generalised symmetric groups, thereby showing how they arise naturally in 2d gauge theory (i.e. the theory of meromorphic connections on vector bundles on curves). This enables us to write down the dimensions of the (global) moduli spaces of rank n, trace-free wild Riemann surfaces for any n, a generalisation of "Riemann's count".