Your mum will need to sign a letter explaining why she's chucking you out, and that you need to claim JSA. Because you're only 16 you may be put into some sort of social care system, you're not old enough to go into 'care' but you may be put into a home where young teens from care would live in once they've hit 16. These tend to be bedsit styled flats with qualified care workers available 24/7 to support the young adults to different extents...some may need help cooking/cleaning; some may just need help finding a job; and some may live completely independently. Usually you'll be placed there but it does take time as they're really overcrowded as it is. You may live in a hostel until you're appropriately housed.
Because you have PTSD you may be prioritised, and may also be able to seek some sort of counselling/support services for free from the NHS.
In terms of what you should do now, go down to your local council gateway and ask for a meeting with an advisor. They'll be able to set up an action plan. Make sure you have all of your documentation like birth certificates etc, as they'll need that to back up your application. Make sure you have letters from your mum detailing the situation so they can also put that on your file.
Also don't listen to people criticising on this forum...I have no idea if you're a troll or not but it's an awful situation to be in hence my advice. As a sister of 2 girls who had to go through the same process (not due to PTSD or violence, but they were forced to leave by our mum's new abusive partner) it's not nice. You need to look at it as an opportunity to live independently, gain some life skills, and seek help for your PTSD...rather than free money. Life on benefits is NOT easy and your lifestyle will be significantly worse than the one you are accustom to at home. I'd seriously suggest looking into options for counselling & self improvement with the help of your mum before you take such a drastic step out there.