Usually inpatient is reserved for people who are deemed to pose a very real threat to their own life or the lives of other people. It's not something that is dished out like sweets.
Usually if you're not suicidal and you go in to have self inflicted injuries treated they'll patch you up, maybe have a chat with you to check that you'll be ok once you leave, and then discharge you.
If you're suicidal and you go in, once again it doesn't necessarily mean they'll make you go inpatient there and then. Usually they'll assess you and see what the best options are.
Some men aren't looking for anything logical, like money.
They can't be bought, bullied, reasoned or negotiated with.
Some men just want to watch the world burn.
If its just sh, you normally see a triage nurse, who asesses the sh, asked a number of related questions, what you used, why you did it, if you feel depressed, suicidal.The nurse will normally clean the wound, if it is a wound and put a dressing on it. Then you wait out in the waiting room until you are called by a doctor or a nurse. Again they ask questions related; How, Why, what with, are you suicidal? etc. If not then they will patch you again and allow you to go and thats it.
If its an od you will normally be taken straight through to a cubicle, bloods taken, again related questions , normally by a doctor; Why, when, what did you take, how much,etc.
As far as going to the A&E if your suicidal. I would guess you the MH team at the hospital would assess you or the crisis team would ( the crisis team did when i od-ed) Asess the risk eg if you have a plan, with date, etc everything sorted. then from there they may want you to be admitted to a ward or you may go IP. Or they may arrange for the crisis team to see you or home treatment team.
In the UK they wont tell your family if your over 18 unless you life is in danger or someone elses life is.