  In this paper, we construct exact rotating wormholes using Ehlers
solution-generating technique. This is based on the Ernst description of
four-dimensional, stationary, and axially symmetric solutions of the
Einstein-Maxwell theory. We adopt the static Barcel\'o-Visser wormhole derived
from the Einstein-Maxwell-conformal-scalar theory as a seed, and demonstrate,
through the Ernst approach, how to construct two novel geometries of rotating
wormholes. These geometries correspond to the Barcel\'o-Visser wormhole
embedded within a rotating and a magnetic background, respectively. In the
first case, the rotation is a result of a dragging force (due to the rotating
background) acting on the initial static wormhole, while in the second case it
is caused by the electromagnetic interaction between the electric charge of the
static wormhole and the external magnetic field. We conduct a comprehensive
analysis of the geometric properties of these configurations, and examine the
new features introduced by rotation, such as the emergence of ergoregions.
Recent evidence suggests that incorporating slow rotation can stabilise
wormholes, rendering these exact, fully rotating solutions particularly
appealing.
