I think my view on the points raised regarding potential poor mental health is that at a 19 year old adult, if his mental health is impacting him from getting a job or furthering his education he should be responsible for making efforts to resolve this. If mental health is the issue, he's old enough to contact his GP, join the waiting list for therapy, improve his sleep hygiene, increase his exercise etc. There are self-help books, meditations online, therapy videos etc, he has plenty of spare time to work on this.
Mental health can be improved in many, many ways so if it transpires that his mental health is the cause, he still has to actively try to improve this.
I would explain to him that although he's decided to hand in his notice at his job, you will still expect your £50 monthly rent and that it will be increasing when he turns 20. He has the two options of either furthering his education/qualifications or to get another full time job. Remaining unemployed is not an option. If he pushes back against this hard, you can remove his devices from the wifi until rent has been paid, to show him that he can't get a free ride with everything handed to him.
If he says that his mental health is struggling, inform him that this is something that the GP can deal with and expect him to seek appropriate treatment. What won't help is allowing him to do nothing - my partner's brother is early 20s, pays no rent to his mum, doesn't work, has no qualifications, sleeps in most of the day and because his mum allows him to do this he has no reason or desire to change.