After a simulacrum uses magic jar to gain control of a new body, can that new body regain hitpoints normally? Or is the new body still restricted by the simulacrum's healing restrictions?
A simulacrum can restore HP in the possessed body.
When Magic Jar is cast, the simulacrum's soul enters the new body changing all of its statistics including HP except for mental ability scores and it retains class features.
... game statistics are replaced by the statistics of the creature though you retain your alignment and your Intelligence, Wisdom, and Charisma scores. You retain the benefit of your own class features.
The original simulacrum body does not regain any hit points; it is "formed from ice and snow" and requires expensive material components to repair.