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 is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Your mother in law is not obliged to stay in the state of Ihraam for 40 days in order to wait for your arrival. She has the choice to opt for Hajj Qiraan, Hajj Tamatu' or Hajj Ifraad. If she enters the state of Ihraam with the intention of making Hajj Qiraan or Hajj Ifraad, then she is obliged to stay in the state of Ihraam until the day of Eed and then she may come out of the first phase of the state of Ihraam or completely come out of the state of Ihraam after fulfilling the conditions of coming out of the state of Ihraam.
However, if she enters the state of Ihraam with the intention of performing Hajj Tamatu', then she performs 'Umrah when she reaches Makkah and then comes out of the state of Ihraam after finishing 'Umrah and she is not obliged to wait for you.
Please refer to Fataawa 86848 and 89057 in relation to a woman travelling to Hajj without a Mahram.
Allaah Knows best.