Hasimara is the nearest station to Bhutan.... But as all trains stop at NJP, then NJP can be considered.... Mahananda is still cancelled, if started then Bhagalpur people can board from Naugachia to Hasimara for Bhutan....
Talking about Bangladesh Border, Malda will always be nearest. Though Bangladesh border is hardly 10 km from NJP via Fulbari one can enter Bangladesh.
Nepal...
more... border I think Jaynagar will still be nearest. Another alternative will be Jogbani.