You can't force him/her if he/she doesn't want to see a doctor. There are many potential mental illness patients in Singapore that went undiagnosed, it doesn't mean they are ok, it's just that they refuse to get treated so they continue to live outside of the treatment regime.
The symptoms you're suggesting sound like hallucinations to me, but I am not an expert so I cannot confirm if it's schizo. If one day your loved ones really go berserk and try to harm oneself or another, that's when the police are going to intervene and maybe send them to IMH for observation. So far that's the time proven way to get a mental sufferer who reject treatment to be hospitalised and given a thorough checkup.
For me, I'm a voluntary sufferer. My father was born in an asylum in Malaysia and my paternal grandmother had mental illness for large phases of her life, so I knew that I had faulty genes and I constantly needed clinical attention for my mental disorders. I willingly surrendered to the police and got handcuffed and hospitalised when I was suicidal in 2018. Before that I had begged my parents to take me to IMH A&E but even when that was done the shrink at the A&E didn't detect anything wrong with me so I went untreated.
Mental illness is an invisible disease, we may think that someone has it, but even when we bring him/her to see a specialist it may still go undetected. Not easy la. Since you have encountered someone in your family who might have mental illness, I have no clear cut solution for you other than empathy, compassion, and equanimity. May you cope with your rigours as a caregiver, and perhaps, through your gentleness and guidance you could encourage your loved one(s) that it's alright to see a mental healthcare specialist.
To start somewhere, you could bring him/her to either Geylang or Queenstown polyclinics to get your loved ones seeing a psychiatric doctor and getting a referral. Polyclinics are usually the first point of contact for mental patients, and it's useful because a referral from the polyclinic means that the future bills are heavily subsidised. It's better than walking into SGH or NUH and seeing a specialist straightaway, the bills are no joke without a proper subsidy.
For me, I was brought into IMH by the police, so I go back for followup every nine to 12 months and I pay around $30 for each visit, inclusive of medicine. It's not that expensive thanks to the Singapore Government's healthcare regime.
