Is it permissible to delay assuming the state of ihram for Umrah from the Miqat until Tanim due to work circumstances, and is a fidyah (expiatory offering) required? And can the fidyah be offered on behalf of one's mother as a gift?

Whoever comes to Makkah intending 'Umrah must enter into ihram from theqat. If one enters into ihram after the miqat, a sacrifice (dam) becomes obligatory upon them. However, if they return to the miqat and enter into ihram from there, then nothing is due upon them.

Your mother owes a sacrifice (dam) for entering into ihram from Al-Taneem instead of the miqat.

As for you, if you were hesitant about performing 'Umrah, it is permissible for you not to enter into ihram from the miqat. Then, if you decide to perform 'Umrah, you would enter into ihram from the Al-Hill (the area outside the Haram boundary but within the miqat boundaries). However, if you were determined to perform 'Umrah, then you must enter into ihram from the miqat, and failing to do so obligates a sacrifice (dam) upon you.

If you offer the expiation (fidyah) on behalf of your mother, you must inform her so that she can make the intention, because expiation is not valid without an intention.