  Recent work has shown the existence of a unique nonlinear extension of
electromagnetism which preserves conformal symmetry and allows for the freedom
of duality rotations. Moreover, black holes and gravitational waves have been
found to exist in this nonlinearly extended electrovacuum. We generalise these
dyonic black holes in two major ways: with the relaxation of their horizon
topology and with the inclusion of magnetic mass. Motivated by recent attention
to traversable wormholes, we use this new family of Taub-NUT spaces to
construct AdS wormholes. We explore some thermodynamic features by using a
semi-classical approach. Our results show that a phase transition between the
nut and bolt configurations arises in a similar way to the Maxwellian case.
