What you are asking to do is create a 'common neutral' circuit where a bunch of different 'hot' conductors share a single neutral conductor. Note, this is not a MWBC, but something even more general. You could have a large number of 'hots' on different or the same phases, all sharing a neutral that was sized for the largest possible unbalanced load. This _might_ have been legal in the past, but is most likely not practical even if there were some set of hoops you could jump through to do a legal install.
In the past these sort of common neutral circuits were explicitly permitted for _outside_ branch circuits, but were not mentioned at all in article 210 for inside branch circuits, so they _might_ have been legal.
It is a requirement that the neutral for a circuit be in the same raceway or conduit as the circuit conductors. So you couldn't go from your new subpanel to your loads and then have the neutral return to the old subpanel. All of the circuit hot conductors and this hypothetical common neutral would go from the new subpanel to the old subpanel in a single raceway, then the common neutral would 'branch out' at the old subpanel. This would probably mean a bunch of derating issues unless there were not many circuits.
I believe that under current rules for MWBCs, you would be forced to put some sort of handle tie on all breakers which share a neutral. This would again make a common neutral solution impractical for more than a few circuits.
I recall that there was a code change prohibiting common neutral circuits in outdoor applications. Not sure if this was explicit for indoor applications. But again even if such circuits are legal, code requirements probably make them impractical.
You are probably better off simply installing the new subpanel at the correct location, and then running new circuits from the new location to the loads.
-Jon