I really don't think we have to posit nanoassemblers for this particular scenario to work. Robot drones are needed, but I think they fall out as a consequence of currently existing robots and the all-singing all-dancing AI we've imagined in the first place. There are shedloads of robots around at the moment - the OP mentioned the existence of Internet-connected robot-controlled cars, but there are plenty of others, including most high tech manufacturer. Sure, those robots aren't autonomous, but they don't need to be if we've assumed an all-singing all-dancing AI in the first place. I think that might be enough to keep the power and comms on in a few select areas with a bit of careful planning.

Rebuilding/restarting enough infrastructure to be able to make new and better CPUs (and new and better robot extensions of the AI) would take an awfully long time, granted, but the AI is free of human threat at that point.