  We study the left-orderability of the fundamental groups of cyclic branched
covers of links which admit co-oriented taut foliations. In particular we do
this for cyclic branched covers of fibred knots in integer homology $3$-spheres
and cyclic branched covers of closed braids. The latter allows us to complete
the proof of the L-space conjecture for closed, connected, orientable,
irreducible $3$-manifolds containing a genus $1$ fibred knot. We also prove
that the universal abelian cover of a manifold obtained by generic Dehn surgery
on a hyperbolic fibred knot in an integer homology $3$-sphere admits a
co-oriented taut foliation and has left-orderable fundamental group, even if
the surgered manifold does not, and that the same holds for many branched
covers of satellite knots with braided patterns. A key fact used in our proofs
is that the Euler class of a universal circle representation associated to a
co-oriented taut foliation coincides with the Euler class of the foliation's
tangent bundle. Though known to experts, no proof of this important result has
appeared in the literature. We provide such a proof in the paper.
