One possible argument is to compare this with $T/C(Z(U))$ as follows. We have $$C(E)/C(E)\cap C(Z(U))\cong C(E)C(Z(U))/C(Z(U)) \leqslant T/C(Z(U))$$  Now $T/U$ is cyclic, so $T/C(Z(U))$ is cyclic. It acts faithfully on $Z(U)$ by conjugation, and if $x$ is an element of $T$ whose image is a generator of $T/C(Z(U))$ then $[x,Z(U)]\leqslant T'$ (since $T/T'$ is abelian), so $x^p$ centralises $Z(U)$ since $T'$ has order $p$. Thus $x^p\in C(Z(U))$, which implies that $T/C(Z(U))$ has order $1$ or $p$.