I think it will depend a little on what country your character is in and the severity of the health issue. Since I live in the UK - and have done for my whole life - I can only give you advice based on our NHS procedures. We do have private hospitals of course, but none I’ve ever had the privilege to use.
My source for this is a family member who has worked in hospitals for a large portion of their life.
The first port of call generally is the Accident & Emergency ward. One thing we need to establish is whether or not your character has been unconscious for days before admittance, or falls unconscious and remains in that state for a few days after they were admitted into A&E. I’ll separate this advice out to make it easier for you to jump to whichever part is relevant.
Unconscious for days in hospital
So, your character has been brought to A&E unconscious by ambulance or whoever found them in that state.
It’s important here to know whether the character has a loved one/relative/friend with them at the time, or whether they are completely alone. If they have someone with them who visits regularly during the period of unconsciousness, then investigations into next of kin will be much easier for the hospital to carry out. Otherwise, the hospital will have to use whatever personal items the patient has with them to try and establish any points of contact/next of kin. We’ll get to the importance of finding a next of kin in a moment.
Anyway, your character is on the A&E ward, trolley-ed up, and will be seen - in an NHS hospital - depending on their priority in relation to other patients who are already on the ward. Since the character is unconscious, they would be considered a high priority - the staff will need to know if your character can a) breathe unaided, and/or b) is suffering any kind of severe head injury/brain damage.
The type of tests that might be carried out on the character include blood tests, x-rays and brain scans to establish why they’re unconscious and what can be done to get the patient responsive again. An A&E ward generally isn’t equipped to carry out brain scans and x-rays, so the patient would need to be taken on their trolley by a porter to wherever they’d need to be.
Bear in mind that paperwork is a huge thing for hospitals - everything that happens to this patient will be documented in several places, and goes through a hierarchy of people. So to keep it basic, things run through the doctor here. They will decide what needs to be done for the patient, the nurses carry out these duties, for example: the doctor decides a blood test needs to be done, the nurse takes the blood and sends it off, then receives the results and passes them onto the doctor.
Anyone admitted to a hospital is always seen by a doctor - nurses don’t make the decisions here.
So after all of this has happened, your character wakes up and they need to get out of their pronto. Let’s say for argument’s sake, they wait for a nurse to come and attend to them. For the moment, we’ll leave out the idea that they’re on a drip or hooked up to monitors.
Generally, hospitals can’t force you to stay under their care, but if you do insist upon leaving, you have to sign a form which basically says that you take full responsibility for yourself once you walk out of the doors. If there is a troubling problem with your character, then it would be likely for doctors/nurses to at least try and persuade them to stay if they believe they need the care and attention.
If the person is adamant that they’ll be leaving, and their tests are generally okay and not anything major, the paperwork procedure can take at least a few hours. The doctor will also likely try to initiate some sort of follow up, such as requesting the character to return for an appointment on x-date at x-time.
In this case, it is possible the character could leave on the exact day they wake up, but the paperwork procedure would be lengthy and if there are any serious concerns, the doctor will try their very best to get the patient to reconsider their choice to leave.
Unconscious for days before going to hospital
So for this example, your character has already been unconscious for a while and for whatever reason has only made it to A&E in time for them to wake up. The same kind of tests might still be carried out, but if, for example, blood tests and scans came back okay without any immediate problems to be seen, the character can self-discharge. This also requires a form of course, and it depends on which tests have been carried out and whether or not the doctor feels it’s okay to let them go.
Again, it is possible the character can leave on the very day they awake. It might be a little quicker than the previous scenario, but not by much I would imagine.
As for next of kin, it’s important because some medical procedures (namely operations) require a form of consent. If your character is unable to give that consent, the hospital would then look for the next available person who could give it to them (if they felt the procedure absolutely integral to the patient’s health). Naturally, any type of complicated procedure or operation would only add time onto the character’s stay, as they would then have to recover from anesthetic, etc.
Tl;dr your character might be able to get out on the day they wake up, but it would take several hours before a doctor would allow it.
Leaving without following procedure
Say the character knows waiting for a nurse/doctor will only impede them and they decide to leave of their own accord… It is possible… but probably not very easy to go unseen. The nurses/doctors on that ward would know of your character as a patient, and even though they likely wouldn’t have time to stand guard over them, they wouldn’t turn a blind eye if they saw this person wandering through the corridors unaided.
Also, a good thing to remember: if your character has been fitted with an IV, yanking it out of their arm will cause a significant amount of bleeding… so they would essentially be wandering the wards, bloodied and maybe a little disorientated in a hospital gown, which would alert a passing medical professional straight away.
Additionally, if they have been hooked up to any kind of monitors, they will react loudly to being forcibly removed, which would alert the attention of the staff on the wards. Leaving during the day would probably be more difficult than leaving during the night, as the wards will be busier and lots of people, including patients, are awake.
In an NHS hospital, the patient - after their brief stay in A&E - would be transferred to a ward with other patients on it. In a private hospital, it is likely they might get their own room, but this all depends on how many beds are available, etc.
So leaving secretly or on the day they wake is entirely feasible, but not without its challenges, Anon…!
Note that all of this information here is very vague… if possible, try and find someone who might have more in-depth knowledge about this, or read books around the subject where you can. Someone else on Tumblr might also see this and correct where I’m wrong, or add in their own two cents, so it’s worth keeping an eye on the notes section!
I hope this helps. Best of luck.