Hi, sorry your family is going through such a tough time. I support DH who has episodes of psychosis. A few years ago after his worst crisis we seperated for a while and he ended up on the streets, so I can draw on that experience.
Firstly is he a british citizen? Secondly how long has he lived abroad as this may impact on what he is entitles to. (I know of a case where someone who had lived abroad for many years was taken ill in the UK and despite being british was told that they had to be resident in UK for 6 months before they were entitled to any SS support including housing. It would probably be worth checking this out with Citizens Advice )
If he is entitled to help he will need to present at the local housing office to explain his situation. This does not necesarily mean that they will be able to provide him with emergency accommodation that day, he may well be turned back onto the streets again (as happened to DH) and have to repeat this process daily until they can find him somewhere (this will likely be hostel or B&B accommodation) He will be awarded a banding (priority level) and can then apply for perminant accommodation, however depending on his banding this could take months. He neds to take proof of his mental illness with him as this will move him higher up the priority list. DH was on the lowest banding as a single male, but was moved to nearly highest once his MH team went to the housing office with him, as he was then classed as a vulnerable adult due to his mental illness.
Getting CMHT support again can be tricky without an address as he would access this through his GP, however if he is no longer registered with one he would need to provide proof of address in order to re register! Was he under a mental health team before he went abroad? It might be worth contacting them and seeing if they would be able to pick him up again or if they would need a new referal. DH was under crisis team before he became homeless, he moved out of area for a few weeks but they were able to pick him up as soon as he returned, this was only a few weeks though. He may be able to self refer to MH services in your area, but not sure, it would be worth looking into.
As for other benefits if he has a sick note from GP/psychiatrist and has been a UK tax payer/contributed national insurance he would be entitled to contribution based ESA, there may be a wait for this though, DH relied on a food bank to bridge this gap. If he has not contributed enough in NI to quakify for this there is a means tested version of ESA which would also require a sick note. He may also qualify for PIP (personal independance payments) though this takes a number of months to set up and he would need to go through an assessment process for this.
Really its a very complicated process made more so by the fact he is living abroad and I'm no expert, can only go on DHs experiences. I would suggest CAB for more reliable advice.