All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Before answering the question, we draw your attention to the fact that if this Hajj to which your wife was given a permit is the obligatory Hajj and she has the ability to perform it, it would not be permissible for her to forsake this opportunity to your mother. Hajj becomes an immediate obligation once a Muslim can afford it, according to the preponderant opinion of scholars, and the Muslim who can afford it is not allowed to delay it without a valid excuse.
As for your question, if the Muslim is capable of performing Hajj through any means and this Hajj has met the conditions of its validity, it will be valid. We do not know about the laws which regulate Hajj affairs where you live, but some laws are laid down to fulfill people’s interests and regulate their affairs, therefore they must be put into consideration and should not be violated.
Allaah Knows best.