what will be the effect (other than no flowers next year perhaps) of cutting a neglected Hydrangea back?

If you want to keep the hydrangea, a hard prune now is not advisable. They are prone to being killed from the top shoots down by cold in the winter. You could just take off a minimal amount if its in the way, and then hard prune when you see growth starting the spring. You will, as you say, probably not get flowers next yearif you hard prune, but it should recover and flower the following year.