  The notion of nonlocal $H$-convergence is extended to domains with nontrivial
topology, that is, domains with non-vanishing harmonic Dirichlet and/or Neumann
fields. If the space of harmonic Dirichlet (or Neumann) fields is
infinite-dimensional, there is an abundance of choice of pairwise incomparable
topologies generalising the one for topologically trivial $\Omega$. It will be
demonstrated that if the domain satisfies the Maxwell's compactness property
the corresponding natural version of the corresponding (generalised) nonlocal
$H$-convergence topology has no such ambiguity. Moreover, on multiplication
operators the nonlocal $H$-topology coincides with the one induced by (local)
$H$-convergence introduced by Murat and Tartar. The topology is used to obtain
nonlocal homogenisation results including convergence of the associated energy
for electrostatics. The derived techniques prove useful to deduce a new
compactness criterion relevant for nonlinear static Maxwell problems.
