Due to the nature of the Academic Mentor-student relationship, students who trust you may decide to disclose what's happening to them before they speak to a doctor or counsellor.
Do offer to help with any potential issues related to your students’ studies and be supportive by listening if you can. You are not expected to, nor should you, take on the role of therapist as this would not be in the best interests of you or your student.
You can recommend that your student thinks about seeking support from a professional such as a GP or the LSE Counselling Services.
You can remind them that this is not because you don't care, but because GPs and counsellors are trained to help and can recommend alternative coping strategies, treatment options and support structures in a way that you're not qualified to do.
If students have not yet registered with a GP, they can do so here. They should also be encouraged to speak to, and register with The Disability and Wellbeing Service who can advise on reasonable adjustments and consider appointments with a Mental Health Adviser if necessary.
Some students may feel uncomfortable visiting a GP or counsellor for fear of judgement.
You should try to reassure them that this is not the case. If students are concerned, reassure them that in the UK medical system, GPs and counsellors will not disclose medical visits to family, partners or teachers without their permission and in most cases, patients are not required to follow any treatment programme counsellors or doctors suggest.
There is a lot more advice and support on what to do when you're concerned about a student here. The Cause for Concern 2019/20 policy and guidelines from the DWS should also be consulted.
In an emergency, ALWAYS call 999.