We consider the contact process on a countable-infinite and connected graph of bounded degree. For this process started from the upper invariant measure, we prove certain uniform mixing properties under the assumption that the infection parameter is sufficiently large. In particular, we show that the projection of such a process onto a finite subset forms a process which is $φ$-mixing. The proof of this is based on large deviation estimates for the spread of an infection and general correlation inequalities. In the special case of the contact process on $\mathbb{Z}^d$, $d\geq1$, we furthermore prove the cutoff phenomenon, valid in the entire supercritical regime.