Yes, but the objectionable current will flow on path of lowest impedance. If the neutral was lost between the new meter and the old meter base, the low impedance pathway for the objectionable current would be the conduit coming out of the top of the old meter base, not the 'nipple' to the panel. (Quotes because isn't actually a nipple).
This does not seem at all likely unless there is also a (code violating) neutral to ground bond in the panel. Otherwise if the neutral was lost downstream of the old meter base, you'd simply have an open neutral, which doesn't seem to be the case here.
Given what we already know about this site, I guess I would not be totally surprised to see some EGCs landed together with neutrals on a single busbar in the panel. That could also explain how current could be flowing on the 'nipple', without a ground-fault per se, and without the panel enclosure being directly bonded to the neutral.
It's also possible there could be another mysterious pathway back to ground through the panel, but we've no reason to believe that yet.