Let $Φ$ a locally convex space and $Ψ$ be a quasi-complete, bornological, nuclear space (like spaces of smooth functions and distributions) with dual spaces $Φ'$ and $Ψ'$. In this work we introduce sufficient conditions for time regularity properties of the $Ψ'$-valued stochastic convolution $\int_{0}^{t} \int_{U} S(t-r)'R(r,u) M(dr,du)$, $t \in [0,T]$, where $(S(t): t \geq 0)$ is a $C_{0}$-semigroup on $Ψ$, $R(r,ω,u)$ is a suitable operator form $Φ'$ into $Ψ'$ and $M$ is a cylindrical-martingale valued measure on $Φ'$. Our result is latter applied to study time regularity of solutions to $Ψ'$-valued stochastic evolutions equations.