  In this paper, we introduce a notion of transfinite nuclear dimension for
$C^*$-algebras, which coincides with the nuclear dimension when taking values
in natural numbers. We use it to characterise a stronger form of having nuclear
dimension at most $\omega$ and moreover, we show that the transfinite nuclear
dimension of a uniform Roe algebra is bounded by the transfinite asymptotic
dimension of the underlying space. Hence we obtain that the uniform Roe algebra
for spaces with asymptotic property C has the corona factorisation property.
