I asked this question on cstheory.se before, where someone pointed out that it is equivalent to asking whether P=NP implies NL=P (thus I edited the question accordingly).
However, my supervisor claims that actually, connectivity being NP-complete implies that NL=P=NP. I can't see why this would be the case; the only additional information we're assuming is that there is a polynomial-time reduction from any NP problem to an NL problem, which does not imply that there exists a corresponding log-space reduction. Wikipedia also mentions
Whether under these types of reductions the definition of NP-complete changes is still an open problem. All currently known NP-complete problems are NP-complete under log space reductions.
... which seems to support my argument.