Who leads mental health: H&S or HR? I think this question comes down to whether you are being proactive or reactive.
In an ideal world there world be a symbiotic relationship between HR and HSE, after all, both focus on people.
On the proactive side of things, HR and Organisational development teams can lead well-being programs. HSE, Occ Health Nurses and even hygienists can be woven into a combined approach. However, the ability of these programmes to drive long term change in physical and mental well-being has been widely debated
Also, smaller organisations don't have the inhouse resources to develop and deploy this kind of programme. Workers are lucky of they have access to an employee assistance programme, workplace medicals and maybe a smoking cessation program and/or paid entry to a fun run.
In the reactive world both HR and HSE are all to often woefully under prepared to deal with burnout, bullying and memtal health issuses for both individuals and the team they work in.
Trying to treat peope with dignity, get them the support they need and support them when they return to work looks simple enough on paper. However, from my experience it's an emotional minefield in real life.
Identifying the complex web of causation and how the workplace contributed to the outcome, identifying how the workplace 'ecosystem' needs to change if it was causativre and ensure that those 'changes' stick is in my opinion a Herculanian task.
Also, if the event goes legal, it will all depend on whetther an employment lawyer chose to challenge the workplace circumstances under the ERA, HSAW or Hunan Rights Act. The retribution factor will enevitably determine which team is required to 'jump' into action after the damage has been done.
Happy to be proven wrong, or questioned on anything I've said.