Am I obligated to offer expiation for an oath if my mother performs 'Umrah after I pledged to the Ministry of Foreign Affairs that the purpose of the visit is solely familial and not for Hajj or 'Umrah?

Your commitment that the visit will be for family only obligates you to fulfill it, as Allah says: "And fulfill [every] commitment. Indeed, the commitment is ever [that about which one will be] questioned" (Quran 17:34), and as the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions." However, if you violate the commitment and your mother performs 'Umrah, you are not required to offer an expiation for an oath (kaffarat yamin) because this commitment is not an oath.