Can a person get long-term nausea from a leg amputation?

No. Unless there is an ongoing issue with the amputation site (infection most likely) nausea is not related to it. Even with active issues nausea would be very rare.
From meds? It is not common or likely to get nausea from a leg amputation itself. It is more likely to get nausea as a side effect from medications being taken.
Unlikely. A well healed amputation should not lead to gastrointestinal symptoms. However , chronic pain, which may present after surgery can be associated with nausea.