  Working within the recent formalism of Poisson-Riemannian geometry, we
completely solve the case of generic spherically symmetric metric and
spherically symmetric Poisson-bracket to find a unique answer for the quantum
differential calculus, quantum metric and quantum Levi-Civita connection at
semiclassical order $O(\lambda)$. Here $\lambda$ is the deformation parameter,
plausibly the Planck scale. We find that $r,t,dr,dt$ are all forced to be
central, i.e. undeformed at order $\lambda$, while for each value of $r,t$ we
are forced to have a fuzzy sphere of radius $r$ with a unique differential
calculus which is necessarily nonassociative at order $\lambda^2$. We give the
spherically symmetric quantisation of the FLRW cosmology in detail and also
recover a previous analysis for the Schwarzschild black hole, now showing that
the quantum Ricci tensor for the latter vanishes at order $\lambda$. The
quantum Laplace-Beltrami operator for spherically symmetric models turns out to
be undeformed at order $\lambda$ while more generally in Poisson-Riemannian
geometry we show that it deforms to \[ \square f+{\lambda\over
2}\omega^{\alpha\beta}({\rm
Ric}^\gamma{}_\alpha-S^\gamma{}_{;\alpha})(\widehat\nabla_\beta d f)_\gamma +
O(\lambda^2)\] in terms of the classical Levi-Civita connection
$\widehat\nabla$, the contorsion tensor $S$, the Poisson-bivector $\omega$ and
the Ricci curvature of the Poisson-connection that controls the quantum
differential structure. The Majid-Ruegg spacetime $[x,t]=\lambda x$ with its
standard calculus and unique quantum metric provides an example with nontrivial
correction to the Laplacian at order $\lambda$.
