  All the magnetically charged ultrastatic and spherically symmetric spacetime
solutions in the framework of linear/nonlinear electrodynamics, with an
arbitrary electromagnetic Lagrangian density $\mathcal{L}(\mathcal{F})$
depending only of the electromagnetic invariant
$\mathcal{F}\!=\!F_{\alpha\beta}F^{\alpha\beta}\!/4$, minimally coupled to
Einstein-scalar-Gauss-Bonnet gravity [EsGB-$\mathcal{L}(\mathcal{F})$], are
found. We also show that a magnetically charged ultrastatic and spherically
symmetric EsGB-$\mathcal{L}(\mathcal{F})$ solution with invariant $\mathcal{F}$
having a strict global maximum value $\mathcal{F}_{_{0}}$ in the entire domain
of the solution, and such that
$\mathcal{L}_{_{0}}=\mathcal{L}(\mathcal{F}_{_{0}})>0$, can be interpreted as
an ultrastatic wormhole spacetime geometry with throat radius determined by the
scalar charge and the quantity $\mathcal{L}_{_{0}}$. We provide some examples,
including Maxwell's theory of electrodynamics (linear electrodynamics)
$\mathcal{L}_{_{_{\mathrm{LED}}}} \!=\! \mathcal{F}$, producing the magnetic
dual of the purely electric Ellis-Bronnikov EsGB Maxwell wormhole derived in
[P. Ca\~nate, J. Sultana, D. Kazanas, Phys. Rev. D {\bf100}, 064007 (2019)];
and the nonlinear electrodynamics (NLED) models given by Born-Infeld
$\mathcal{L}_{_{_{\mathrm{BI}}}} \!=\! -4\beta^{2} + 4\beta^{2} \sqrt{ 1 +
\mathcal{F}\!/\!(2\beta^{2})~}$, and Euler-Heisenberg in the approximation of
the weak-field limit $\mathcal{L}_{_{_{\mathrm{EH}}}} \!=\!
\mathcal{L}_{_{_{\mathrm{LED}}}} + \gamma \mathcal{F}^{2}\!/2$. With those NLED
models, two novel magnetically charged ultrastatic traversable wormholes (EsGB
Born-Infeld and EsGB Euler-Heisenberg wormholes) are presented as exact
solutions without exotic matter in EsGB-$\mathcal{L}(\mathcal{F})$ gravity.
