Severe voltmeter needle oscillations

Carburetted vehicles had a voltage regulator that controlled the field current. Some controlled the volts to the fiel and some controlled the ground.
You can measure the voltage to the regulator and with a DMM the Hz or frequency.
Your problem could be the regulator, but I would suspect worn brushes. Worn brushes have low spring force holding the brushes to the slip rings. Any out of concentric or also probably present loose bearings allows the armature to jump around as the revs come up, causing the brushes to make and lose contact. This is RPM related.