  We present results for the hadronic running of the electromagnetic coupling
and the weak mixing angle from simulations of lattice QCD with $N_f=2+1$
flavours of $O(a)$-improved Wilson fermions. Using two different
discretisations of the vector current, we compute the quark-connected and
-disconnected contributions to the hadronic vacuum polarisation (HVP) functions
$\bar{\Pi}^{\gamma\gamma}$ and $\bar{\Pi}^{Z\gamma}$ for spacelike squared
momenta $Q^2\leq 7$ $\mathrm{GeV}^2$. Our results are extrapolated to the
physical point using ensembles at four lattice spacings, with pion masses
ranging from 130 to 420 MeV. We observe a tension of up to 3.5 standard
deviations between our lattice results for $\Delta\alpha_{\rm had}^{(5)}(-Q^2)$
and estimates based on the $\textit{R}$-ratio for space-like momenta in the
range $Q^2=3-7\,\rm GeV^2$. To obtain an estimate for
$\Delta\alpha_\mathrm{had}^{(5)}(M_Z^2)$, we employ the Euclidean split
technique. The implications for comparison with global electroweak fits are
assessed.
