I reached out to my university's wellbeing team as soon as I started. I contacted the team via email and was invited to book a face to face assessment. My university has a Risk Register, which is a register of students they are concerned about or who could benefit from additional support. These students are allocated a member of staff from the Wellbeing Team, who acts similarly to a Care Coordinator. They contact the student weekly to check in with them and support them, and based on the student's needs, the team can provide extra support such as more 1:1 sessions, access to the uni's gym, putting them in touch with other students, etc. The Wellbeing team also liaises with teaching staff to inform them of any issues students are having, with permission. I had regular face to face appointments, and check-ins via telephone. If I ever needed a last minute appointment, I could walk in and ask to speak to someone from the team, or email and arrange an appointment if I had more notice.
During my first year of university, I experienced a serious event and the Wellbeing team were the first people I told. Their support was incredible, and life saving. They supported me through reporting it to the police, through the entire investigation, through having to take time out from my degree. I had far more contact than normal, in that the team kept checking in and things for over a year. Sometimes calls would be fortnightly or longer, sometimes they were more frequent if needed. I feel my uni's Wellbeing team work incredibly hard to do everything they can to help support students and their support was invaluable to me. I felt the most supported I had ever been in my life.
I’d always encourage students to talk to their uni's Wellbeing and Student Support teams. I have been to three different universities and the support has varied, but it has definitely improved in recent years. Student support services are separate to teaching staff, and so info you tell them is unlikely to be shared. You also cannot be kicked out of uni for struggling. It is far, far better to reach out and ask for help. The help is also often quicker through uni, to access, i.e. counselling, than it is via the NHS.