  Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz,
Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on
Cantor set-like constructions using nested intervals. We exhibit a class of
examples in non-autonomous iteration where one considers compositions of
polynomials from a sequence which is in general allowed to vary. In particular,
we give a sharp criterion for when Julia sets from our class will be HNUP and
we show that the maximum possible Hausdorff dimension of $1$ for these Julia
sets can be attained. The proof of the latter considers the Julia set as the
limit set of a non-autonomous conformal iterated function system and we
calculate the Hausdorff dimension using a version of Bowen's formula given in
the paper by Rempe-Gillen and Urb\'{a}nski \cite{RU}.
