If you need a reference, there is a paper that covers your question.

Varadarajan (1958): If $\Omega$ is a separable metric space, then the topology of weak convergence on $\mathcal{M}(\Omega)$ is metrizable if and only if the weak convergence and norm topologies on $\mathcal{M}(\Omega)$ coincide.

This condition is obviously violated if $\Omega$ is the unit interval (or any other uncountable separable metric space).

The same paper also shows that the weak convergence topology on the subset $\mathcal{M}^+(\Omega)$ of positive measures is metrizable.

If you need a reference, there is a paper that covers your question.

Varadarajan (1958): If $\Omega$ is a separable metric space, then the topology of weak convergence on $\mathcal{M}(\Omega)$ is metrizable if and only if the weak convergence and norm topologies on $\mathcal{M}(\Omega)$ coincide.

This condition is obviously violated if $\Omega$ is the unit interval (or any other uncountable separable metric space).

The same paper also shows that the weak convergence topology on the subset $\mathcal{M}^+(\Omega)$ of positive measures is metrizable.