Some of the attitudes on this thread truly horrify me. If you are able in any way to help him I hope you can.
Yes at the moment he is a 19 year old NEET but he is also a teenager who lost his mum unexpectedly 20 weeks ago and his whole world has been turned upside down.
I know in the past many young men his age married, had families and even went to war but society has dramatically changed along with the realisation of medical fact that young adults aren't fully matured mentally until their early 20's (even the prison system recognises this). As such we live in a time where we now treat young adults differently, some people may even say we infantilize young people too much but this is not his personal fault it is just how our society in this country has evolved.
This young man needs love, support and firm guidance. His current situation is not sustainable but with help and direction he can make changes that will hopefully set him up to lead a successful, productive life. He is at a critical stage in his life where he may either sink of swim.
Along with a roof over his head I would also recommend that he is offered the chance to access grief counseling if he wants it.
Clear boundaries need to be set with a understanding of what your expectations are. Whether he chooses to re enter education or get a job he needs to realise that some financial contribution is needed for practical reasons. He needs to temporarily seek benefits so he is able to help pay his way but we all know even this will take time initially.
It may take him a few months to get sorted but with your support he will find this immeasurably easier. If he wishes to go back into education this may not be possible until September so he will need to actively seek employment until then.
If he is unwilling to seek education or employment then he needs to be made aware that this is unacceptable long term and only offer your home on a fixed short term basis. Tell him that you love him and will help him get back on his feet but it will require effort and determination from him and a promise that your home and family will be respected. He has reached out to you and how you respond could quite literally shape his future.