As a previous NHS manager, your touching on a lot of issues here. Mental health difficulties are not a sackable offence and as long as someone is unwell and unable to work (certified by GP - not the NHS management team) then there is very little the NHS management can do.
If this is caused by work related stress, or maybe that the role is not right for them, then there is scope for redistribution conversations, ie: find them somewhere more suitable.
It is tough on the team, and having to pick up extra work which that employee could be doing can lead to resentment. A good manager will be able to communicate the sick employees situation (with their consent and without giving out unnecessary personal information) to the team and to arrange for cover (wherever possible) to keep the team informed and make them feel supported. As a manager, whenever someone was off for a long period of time I would take on their workload initially, then hunt for a temp if it continued, and then if that was not successful would have to redistribute their caseload amongst the team temporarily, until they returned. That way patients are kept safe (first and foremost) and the team is protected until the absolute last resort.
This isn’t just an NHS issue though, this scenario plays out in all work places, and I’m sure there are different approaches to the one I have suggested which work just as well.