What is the ruling on giving out one's first salary for the sake of Allah, given a subsequent increase in salary and the ability to pay after a period of time, and is it sufficient to give it for the purpose of helping one's mother perform 'Umrah?

This is a temporary vow that must be fulfilled immediately by spending the first salary you received. You must repent for delaying the fulfillment of the vow without a valid excuse.

You are not obligated to offer expiation according to the most correct opinion (the view of the majority of scholars), but there is no harm if you offer it as a precaution.

It is not permissible to give the vowed amount to your mother for Umrah; instead, it must be spent on the poor and the needy.