Equivalently you can also define $\theta:=\Theta( p)$ as the smallest increasing function on $I:=[0,1]$ which is larger than $p$ on $I$ (then your definition would rather be the construction of $\theta$). Also note that you can define this inferior envelope of increasing functions for any bounded function $p$ on $I$.
A simple fact: by the construction, for any $0\le s \le t\le 1$ there are $ s\le s'\le t'\le t $ such that $p(t')=\theta(t)$ and $p(s')=\theta(s) $. As a consequence, if $p$ is continuous and $\omega$ is a(n increasing) modulus of continuity for $p$, then $\omega$ is also a modulus of continuity for $\theta$. In particular, if $p$ is $k$-Lipschitz so is $\theta$, and if $p$ is $\lambda$-Hölder, so is $\theta$, with $\|\theta\|_\lambda\le\|p\|_\lambda$ .
(For a polynomial $p$, one can also observe that $\theta$ is a piecewise $C^1$ function, hence Lipschitz. In fact $\|\theta'\|_{\infty,[0,1]}\le\| p'\|_{\infty,[0,1]}$, because either $t$ is a local maximum of $p$, or $\theta'(t)=0$ or $\theta'(t)=p'(t)$, so $\rm{Lip}(\theta)\le \rm{Lip}( p)$).