Who leads mental health: H&S or HR? The two places I worked at prior to my current position weren't interested in applying anything to do with mental health/wellbeing practices, but if anything along those lines did raise its head, it was either thrown at the H&S person (me) or fobbed off to the EAP.
Unless it was something like bullying or harassment, then (in the first prior job) this was taken over by HR (H&S person told politely - but firmly - "This has nothing to do with you"), which 9 time out of 10 resulted in the complainant usually leaving after being paid out and the accused usually got nothing more than a warning.