Uniform Space whose Topology is Metrizable is not necessarily Metrizable

Theorem
Let $\mathcal U$ be a uniformity on a set $S$.

Let $\left({\left({S, \mathcal U}\right), \tau}\right)$ be the uniform space generated from $\mathcal U$.

Let $T = \left({S, \tau}\right)$ be the uniformizable space yielded by $\left({\left({S, \mathcal U}\right), \tau}\right)$.

Let $T$ be a metrizable space.

Then it is not necessarily the case that $\mathcal U$ is itself a metrizable uniformity.

Proof
Let $T = \left({S, \tau}\right)$ be an uncountable discrete ordinal space.

From Uncountable Discrete Ordinal Space is Metrizable, $T$ is a metrizable space.

However, from Uncountable Discrete Ordinal Space has Unmetrizable Uniformity, there exists a uniformity $\mathcal U$ which yields the uniformizable space $T = \left({S, \tau}\right)$ which is not itself a metrizable uniformity.