If you go to the hospital or call an ambulance or the police to take you there and tell them you feel like you are going to kill yourself they will admit you and send you into a mental health clinic (or refer you to one and then give you a ride there if you need it) where you can be properly diagnosed and get medication.
If you are admitted as suicidal you will usually be kept 72 hours then released if you feel you are ready to leave.
There aren't really any 'mental institutions' like you see in movies anymore. It's more likely to be a building with a couple different large rooms with multiple beds, men in one and women in another, a kitchen/dining room, rec room, and a few meeting rooms. It's not unpleasant at all, you'll get to talk to doctors and nurses who will help figure out what's wrong and what you want to do about it. You will then be able to get extended care through them or someone else.
If you can't afford treatment they will find somewhere that is low or no cost to you. I get treatment free because I don't qualify for assistance and can't afford to pay for it myself.
I think it probably differs here in the UK, because my local hospital (and at least one other I've been to as a visitor) has on-site mental health wards, and since I took my mum in when she relapsed, I know the proceedure.
The possibility of harm wouldn't be the main focus, either. The main focus is that something is wrong and I need help, the harm is a possible consequence of not getting it. As much as it seems like a way of making sure I'm seen, I'd only mention it because it's true; the possibility is there but I want to make it clear that I've come voluntarily in order to avoid that consequence, and to help remedy the situation.
I will undoubtedly reach a stage where I will cause myself and/or others around me a great deal of harm, be it through a violent attempt to escape a situation, or through an attempt on my own life.
Having been an psychiatric in-patient on two separate occasions, I'm not very fond of the process, but the fact that you've admitted this to yourself makes me think you're a good candidate for a psychiatric institution. Even if you're having difficulties with your home life right now, hurting yourself or someone else will make things much, much worse, so you need to do whatever possible to get yourself into a more stable mental state, even if that means checking into the hospital for a couple days. No matter how long they keep you (I don't know what UK policy is, but if it's like in America you probably won't be there more than a few days), it's a blip in time, and if it helps even the slightest, it'll be worth it.
The only downside to hospitalisation is that I will be without everything that has kept me at least slightly sane. My computer and friends (I have no friends currently in close proximity) will not be there, I will be unable to continue my writing outside of notepads, I will have no access to my warhammer models to play with or to paint, and I will not have my bookshelf. Everything I rely on to take my mind off what I'm going through will not be there.
Your feelings are understandable, but I want to point out a few things: 1) Adult psychiatric wards, especially where you are admitted on a voluntary basis, give you a lot more freedom than you might think. When I was admitted, they let me bring books and keep pencils and paper to write with. No, you don't get access to computers and no matter what they allow, it won't be the same as home. But if you have your creature comforts and you're still considering killing yourself, then they're not working anyway.
2) On a similar note, functionally mentally ill people (that is, people who can bathe, feed themselves, etc.) and those who cannot care for themselves/aren't in their proper minds usually are kept separate to some degree. If the hospital and staff is worth their salt, they'll acknowledge your concerns and keep that in mind while they treat you. They don't want you to feel uncomfortable and more stressed because you're worried about someone hurting you. You'll find that many of the people there (I'll even go as far as to say the majority) are just like you - aware and functional to a basic degree, but just dealing with some difficult emotional or environmental stuff. Because they're suffering, they'll be sympathetic to your situation and could actually offer exactly the kind of support you're lacking at home. Both times I was committed, 90 percent of the people were alcoholics, depressed, or bipolar. These were the same kind of people I met on the street or went to school with, not "mad" at all. There was one severely schizophrenic woman who was obviously unable to care for herself, but she was in a completely different ward and I was only in her company for about 10 minutes during dinner one day; the rest of the time was spent in my room or in group therapy with people in similar conditions. So try not to assume you'll be out of place; some people I met were a delight to talk to. Some people even stay in contact as friends once they leave.
3) A psychiatric ward has more flexibility with regard to medical treatment because they can monitor you closely for side effects or improvement, so if your current doctor is unwilling to give you medication and you think it could help, you'll definitely have access to those resources in the hospital. You'll also be able to talk with other professionals who could offer a refreshing change of perspective or course of action.
I would be lying if I said I liked being in a hospital; I would've loved to avoid it if I could have. But when you're at the end of your rope, you have to be willing to do whatever you can (within reason) to get better. Just don't give up.
One of the reasons I've been thinking about admitting myself is that I would be seen by someone more qualified to understand anything I'm suffering from, and that it would be done quickly. To get my current therapist, I had to wait nearly a year, despite being told I'd be seen in 6 months. I'm not complaining about that, but I am worried that going through the referral channels would take too long for preventative measures to even be viable, nevermind effective.
The conditions of psychiatric wards are giving me a bit of confidence, but if the UK is indeed stricter as hippo says, then it might be a false hope, but I might as well treat it as being as you say.
As for access to medication, I've been giving my mum the same cocktail of pills for years, and I'd hate to become dependant on anything so powerful, or as many things as she's dependant on. Medication isn't what I'd go looking for, but I'd be willing to accept it.
That is definitely a good reason to consider hospitalization. I'm going to get a second opinion from a new psychiatrist in December, but I had to make the appointment months ago. Mental health professionals are horrendously overbooked and overworked. At the hospital you would see a psychiatrist and social workers every day instead of every six months, which means you have a much better chance of getting diagnosed and treated properly.
But it's extremely important that you know what you're getting into before you commit to anything. Take a good look at your local and national mental health laws -- they should be public record. Find out whether there's a minimum stay or if you can check yourself out whenever you're ready to leave. Find out the legal definition of involuntary and voluntary hospitalization, find places in your area that offer them and research. Also make SURE you know your rights in detail; professionals are not always as explicit as they should be as far as procedure, so if you choose to be hospitalized (and even if you don't), write down all the questions you have and get a detailed answer to each one. Knowing and demanding your rights could be the major difference from a pleasant experience and a horrible one. Don't take anyone's shit, and don't let anyone force treatment on you that you're not 100 percent sure of. It's your health, so you need to be your biggest advocate no matter what. Do that, and everything else will fall in place. Best of luck.