Here is an example of a "2-step nilpotent" unipotent group $E_1$ for which
the quotient $V=E_1/[E_1,E_1]$ is a vector group (so every element of $V(k)$
is $p$-torsion) but for which not every element of $V(k)$ may be lifted
to a $p$-torsion element of $E_1(k)$.

This confirms BCnrd's skepticism in one of the comments. I admit that this
example may well not arise as a subgroup of $U/[U,[U,U]]$ for the unipotent
radical $U$ of a Borel (though I don't see precisely how to argue that it doesn't).

To form $E_1$, I want to construct an extension of a vector group by the
additive group $\mathbf{G}_a$
using the sum of two 2-cocycles. I'll first describe each of these separately.

First, recall that the additive group of length
2 Witt vectors $W_2$ is a self-extension of $\mathbf{G}_a$
$$0 \to \mathbf{G}_a \to W_2 \to \mathbf{G}_a \to 0$$
defined by a certain 2-cocycle $\sigma$. Addition is given by
$(a_0,a_1) + (b_0,b_1) = (a_0 + b_0,\sigma(a_0,b_0) + a_1 + b_1)$ where
$\sigma(X,Y) = \dfrac{1}{p}(X^p + Y^p - (X+Y)^p) \in \mathbf{Z}[X,Y]$.

Next, let $V$ be a finite dimensional $k$ vector space with $\dim V \ge 2$, viewed as a vector group over $k$.
Let $\beta$ be a non-deg alternating form on $V$. Then $\beta$ defines
a non-commutative central extension
$$1 \to \mathbf{G}_a \to E \to V \to 1$$
which I'll write multiplicatively: the operation will be given by
$(v,a)\cdot(w,b) = (v+w,\beta(v,w) + a + b)$. (I'm identifying
the variety $E$ with $V \times \mathbf{G}_a$).

The extension I want is a hybrid.
Fix a non-zero linear functional $\phi:V \to k$ and consider the extension
$$1 \to \mathbf{G}_a \to E_1 \xrightarrow{\pi} V \to 1$$
with operation given by
$(v,a)(w,b) = (v+w,\sigma(\phi(v),\phi(w)) + \beta(v,w) + a + b)$.

Since $\beta$ is non-degenerate we have $E_1/[E_1,E_1] \simeq V$. If $L \subset V$
is a line for which $\phi(L) \ne 0$, the subgroup
$\pi^{-1}(L) \subset E_1$ is isomorphic to $W_2$. In particular,
any element $(v,a) \in E_1(k)$ with $\phi(v) \ne 0$
has order $p^2$.