NC = P consequences?
Yes. $NC$ can be seen as the class of languages recognized by alternating Turing machines that use $O(\log n)$ space and $(\log n)^{O(1)}$ time. (This was first proved by Ruzzo.) $P$ is the class where alternating Turing machines use $O(\log n)$ space but can take up to $n^{O(1)}$ time. For brevity let's call these classes $ATISP[(\log n)^{O(1)},\log n] = NC$ and $ASPACE[O(\log n)] = P$. 

Suppose the two classes are equal. Replacing the $n$ with $2^n$ in the above (i.e., applying standard translation lemmas), one obtains 

$TIME[2^{O(n)}] = ASPACE[O(n)] = ATISP[n^{O(1)}, n] \subseteq ATIME[n^{O(1)}] = PSPACE$. 

If $TIME[2^{O(n)}] \subseteq PSPACE$ then $EXP = PSPACE$ as well, since there are $EXP$-complete languages in $TIME[2^{O(n)}]$.

Edit: Although the above answer is perhaps more educational, here's a simpler argument: $EXP = PSPACE$ already follows from "$P$ is contained in polylog space" and standard translation. Note "$P$ is contained in polylog space" is a much weaker hypothesis than $NC = P$. 

More details: Since $NC$ circuit families have depth $(\log n)^c$ for some constant, every such circuit family can be evaluated in $O((\log n)^c)$ space. Hence $NC \subseteq \bigcup_{c > 0} SPACE[(\log n)^c]$. So $P = NC$ implies $P \subseteq \bigcup_{c > 0} SPACE[(\log n)^c]$. Applying translation (replacing $n$ with $2^n$) implies $TIME[2^{O(n)}] \subseteq PSPACE$. The existence of an $EXP$-complete language in $TIME[2^{O(n)}]$ finishes the argument.

Update: Addressing Andreas' additional question, I believe it should be possible to prove something like: $EXP=PSPACE$ iff for all $c$, every polynomially sparse language in $n^{O(\log^c n)}$ time is solvable in polylog space. (Being polynomially sparse means that there are at most $poly(n)$ strings of length $n$ in the language, for all $n$.) If true, the proof would probably go along the lines of Hartmanis, Immerman, and Sewelson's proof that $NE = E$ iff every polynomially sparse language in $NP$ is contained in $P$. (Note, $n^{O(\log^c n)}$ time in polylog space is still enough to imply $PSPACE=EXP$.)