  We investigate the existence of Ulrich vector bundles on suitable $3$-fold
scrolls $X_e$ over Hirzebruch surfaces $\mathbb{F}_e$, for any integer $e
\geqslant 0$, which arise as tautological embeddings of projectivization of
very-ample vector bundles on $\mathbb{F}_e$ that are uniform in the sense of
Brosius and Aprodu--Brinzanescu.
  We explicitely describe components of moduli spaces of rank $r \geqslant 1$
vector bundles which are Ulrich with respect to the tautological polarization
on $X_e$ and whose general point is a slope-stable, indecomposable vector
bundle. We moreover determine the dimension of such components, proving also
that they are generically smooth. As a direct consequence of these facts, we
also compute the Ulrich complexity of any such $X_e$ and give an effective
proof of the fact that these $X_e$'s turn out to be geometrically Ulrich wild.
  At last, the machinery developed for $3$--fold scrolls $X_e$ allows us to
deduce Ulrichness results on rank $r \geqslant 1$ vector bundles on
$\mathbb{F}_e$, for any $e \geqslant 0$, with respect to a naturally associated
(very ample) polarization.
