Résumé
We consider the isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, and any polar divisor; allowing for a twisted/ramified formal normal form at each pole, and for twists in the interior of the curve. (This covers the general case in 2-dimensional meromorphic gauge theory.) We focus on the irregular moduli of the connections, studying the fundamental groups of the spaces of admissible deformations of their irregular types/classes, i.e., the local wild mapping class groups in the title. To describe them, we first take the viewpoint of (nonsplit) reflections cosets in Springer/Lehrer--Springer theory, which yields in particular new modular interpretations of complex reflection groups -- and their braid groups. Then we introduce new `fission' trees to treat structure groups of any (simple) classical type, leading to a complete classification of the corresponding hyperplane arrangements, and singling out an infinite family of noncrystallographic examples in type D. Moreover, we reinterpret Bessis' lift of Springer theory as the study of `quasi-generic' deformations, corresponding to irregular singularities whose leading coefficient is regular semisimple upon pullback along a local cyclic covering of the base curve. Finally, we rephrase much of this material in terms of generalized root-valuation stratifications.