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 $\mathcal{F}={F}_{\ensuremath{\alpha}\ensuremath{\beta}}{F}^{\ensuremath{\alpha}\ensuremath{\beta}}/4$, 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 100, 064007 (2019)]; and the nonlinear electrodynamics (NLED) models given by Born-Infeld ${\mathcal{L}}_{\mathrm{BI}}=\ensuremath{-}4{\ensuremath{\beta}}^{2}+4{\ensuremath{\beta}}^{2}\sqrt{1+\mathcal{F}/(2{\ensuremath{\beta}}^{2})}$, and Euler-Heisenberg in the approximation of the weak-field limit ${\mathcal{L}}_{\mathrm{EH}}={\mathcal{L}}_{\mathrm{LED}}+\ensuremath{\gamma}{\mathcal{F}}^{2}/2$. 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, and we show that these solutions have in common the property that in the weak electromagnetic field region the magnetically charged Ellis-Bronnikov EsGB Maxwell wormhole is recuperated.