Well Zuko's mother may never be found out it has been proven that shes alive so here are some possible outcomes in my opinion

1. 1(most possible)

she was exiled to the earth kingdom to never return to the fire nation.

1. 2(possible)

she was sent to the boiling rock for treason and passed away and never seen by zuko when he went there

3# not really possible)

but still a possibility she was banished from the throne and lives in a fire nation town

4#(not possible)

she moved to the water tribe and was the grandmother of tarlock