Implications of $NL=P$
$NL = P$ means that every language in $P$ can be decided by a nondeterministic Turing machine using at most $O(\log n)$ space, where $n$ is the size  of the input.

Savitch's theorem states for all $f(n) \in \Omega(\log n)$: $NSPACE(f(n))\subseteq SPACE(f(n)^2)$.

Therefore $NL = P$ implies that every language in $P$ can be decided by a deterministic Turing machine using at most $O((\log n)^2)$ space.