From a purely electrical (not code) point of view this is pretty simple. A 20A breaker in the new main won't protect any existing #14 final circuits in the old panel. So the old panel needs to have 15A fuses for any #14 final circuits. Then the 20A breaker is OK, if the feeder can take it it could even be 25 or 30A.
Completely different topic: are the electricians required to correct existing overfused circuits if they only hook up the feeder to a different source? Around here the answer would be yes, the electrician doing any work at the place is responsible for violations he should have noticed, providing it has already been a violation when the offending equipment was installed.
