They are important but not life threateningly important.
The awful situation with your Dad's treatment is a separate issue.
In (what is described as) single sex hospital accommodation eg wards, showers then safety, privacy and dignity are incredibly important.
The majority of people in the UK has very good reason to expect that they are singe sex because this is what they have been told.
It matters a lot hence why the policy is a performance indicator.
This can apply to closed psychiatric wards as well as reported:
'Lancaster mum with “fear of men” locked on hospital ward with transgender patient'
(extract)
"A Lancaster mum whose bi-polar disorder left her believing men were conspiring to kill her said she was left terrified when she was locked on a women’s psychiatric ward with an “extremely male-bodied” transgender patient. Philippa Molloy, 42, said she was “genuinely, absolutely terrified” because she had suffered a relapse in her condition that made her irrationally terrified of men – including her own husband.
When she raised her concerns with hospital staff, however, she said she was not taken seriously and her medical notes implied that she was a “transphobic bigot”. She said the NHS had failed to think through the implications of allowing patients to self-identify their gender.
“The rights of that trans person to feel safe were put above the rights of me to feel safe as a natal woman,” she said." (continues)
www.lancasterguardian.co.uk/news/lancaster-mum-with-fear-of-men-locked-on-hospital-ward-with-transgender-patient-1-8963648