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.
Hajj must be performed at once according to the sounder opinion of scholars. Thus, the person who is capable of performing Hajj should perform it at once except if he has an excuse. If the person has an excuse, he may delay Hajj until he becomes able to perform it. Hence, if your sister did not have any one to take care of her during her pregnancy and after her labor, then apparently your mother's stay with her would be an excuse that makes it permissible to delay Hajj to the following year and she would not be held sinful, Allaah Willing.