We construct traverseable wormholes by starting with simple four-dimensional classical solutions respecting the null energy condition and containing a pair of oppositely charged black holes connected by a non-traverseable wormhole. We then consider the perturbative back-reaction of bulk quantum fields in Hartle-Hawking states. Our geometries have zero cosmological constant and are asymptotically flat except for a cosmic string stretching to infinity that is used to hold the black holes apart. Another cosmic string wraps the non-contractible cycle through the wormhole, and its quantum fluctuations provide the negative energy needed for traversability. Our setting is closely related to the non-perturbative construction of Maldacena, Milekhin, and Popov (MMP), but the analysis is complementary. In particular, we consider cases where back-reaction slows, but fails to halt, the collapse of the wormhole interior, so that the wormhole is traverseable only at sufficiently early times. For non-extremal backgrounds, we find the integrated null energy along the horizon of the classical background to be exponentially small, and thus traversability to be exponentially fragile. Nevertheless, if there are no larger perturbations, and for appropriately timed signals, a wormhole with mouths separated by a distance d becomes traverseable with a minimum transit time . Thus is smaller than for the eternally traverseable MMP wormholes by more than a factor of 2, and approaches the value that, at least in higher dimensions, would be the theoretical minimum. For contrast we also briefly consider a ‘cosmological wormhole’ solution where the back-reaction has the opposite sign, so that negative energy from quantum fields makes the wormhole harder to traverse.