  In this paper, we present an abstract framework to obtain convergence rates
for the approximation of random evolution equations corresponding to a random
family of forms determined by finite-dimensional noise. The full discretisation
error in space, time, and randomness is considered, where polynomial chaos
expansion (PCE) is used for the semi-discretisation in randomness. The main
result are regularity conditions on the random forms under which convergence of
polynomial order in randomness is obtained depending on the smoothness of the
coefficients and the Sobolev regularity of the initial value. In space and
time, the same convergence rates as in the deterministic setting are achieved.
To this end, we derive error estimates for vector-valued PCE as well as a
quantified version of the Trotter-Kato theorem for form-induced semigroups.
