Let $Φ'$ denote the strong dual of a nuclear space $Φ$. In this paper we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of $Φ'$-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of $Φ'$-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of linear stochastic evolution equations driven by semimartingale noise.