Let $X$ be a metrizable topological space, and let $U$ be a uniformity which induces the topology on $X$. My question is, is $U$ necessarily metrizable? That is, is $U$ induced by some metric on $X$?
If not, is there an example of a uniformity for a metrizable topology which isn’t induced by any metric?
No ,this is not the case. I had an answer to a similar question here. The example mentioned there: take the reals in the standard topology and let $\mathcal{U}_f$ be the so-called fine uniformity on the reals (the finest uniformity on the set that induces the same topology). Then this uniformity is not metrisable by standard results quoted in this survey paper. Beware, there is a lot of theory behind this (the paper also quotes bornological conditions, which will please the OP).