All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger.
A secret is a trust (Amaanah), and disclosing it is a breach of trust, especially if the person who confided in you asked you to keep the secret. It is impermissible to disclose it; Jaabir narrated that the Prophet, sallallaahu ʻalayhi wa sallam, said, “If a man says something and then turns away, it becomes a trust [which should not be disclosed by the one who heard it].” [Au Daawood and At-Tirmithi] Ibn Sallaam said, “The Prophet, sallallaahu ʻalayhi wa sallam, described it as a trust although the one who heard it was not asked to keep it a secret; how then would it be if he was asked to keep it a secret?” [Ghareeb Al-Hadeeth]
Al-Hasan Al-Basri said, “It is a kind of betrayal that you disclose your brother's secret.”
However, if there is a preponderant benefit, then there is no harm in disclosing it. So if you fear that she (your sister) would incur harm upon herself by taking this loan and you hope that your mother would help you deter her, then there is no harm in telling your mother about it.
If this loan is based on interest, then it is impermissible for your sister to get it and it is pure harm without doubt, even if it is through Muraabahah (cost-plus financing) or similar lawful financial transactions. Although such transactions are legitimate, the Muslim should not incur debt except if there is a valid need. Taking such a huge loan often involves incurring harm upon oneself, such as being unable to repay it.
Allah knows best.