(Warning: long reply is long)
Hi, and welcome to our huge family!

I don't believe we've met yet... so here's a quick potted history: I'm a 43-year-old Sassenach who started transitioning at age 40, after having married & had kids. I've had gender dysphoria since 1976, and I realised I was trans in 1990 but the surgical options back then were so dire that I forced myself back into the closet and tried to live my life as best as I could until the dam finally burst a couple of years ago. As you can see from my tickers, I've been on T for a while & I've had top surgery with Mr Yelland in Brighton. I'm also on the autism spectrum (Asperger's), and I grew up with an incredibly abusive, neglectful family.
Yes, it hurts incredibly to be stuck with an unaccepting family. When I was 15 I couldn't see the light at the end of the tunnel: as far as I was concerned, there
was no light. But just a few years later I moved out of home and started living my life the way I wanted to at the time. Whenever things feel hopeless, just remember that it's only a short time until you can stand on your own two feet. I can tell you that life as a 15-year-old is very different to life as a 20-year-old, which is very different to a 25-year-old. Do you honestly believe that you'll give a hoot what your mother says when you're 25? Of course not.

Very soon, your folks won't have any say in your life.
I see you've mentioned elsewhere that you're Scottish which means you're likely to go to the Sandyford in Glasgow. The good news is that they can give you testosterone from age 16 and they can also put you on hormone blockers if those are required. Also, from age 16 you have the right to consent to (or refuse) medical treatment without your parents' knowledge or permission. So very soon, you'll be able to attend a GIC without your family even knowing about it, if that would help.
You can refer yourself to the Sandyford (you don't even need to see your GP first): just ring them on 0141 211 8137, tell them your age and ask for a first appointment. They have a youth service, although they generally want your parents to be involved if you're under 16 - so explain to them that your family is unsupportive and ask if they can keep things confidential. Worst case scenario would be that they'd ask you to wait until you're 16, but that's just around the corner, isn't it? If you're not in Glasgow, you can claim for the cost of travelling to the clinic. You need all the support you can get: presuming you're still in school, do you have a school counsellor you can talk to? Does your school have an LGBT society you could join? Are there any LGBT youth groups in your area?
It sucks that your family won't buy you a binder, but there are schemes where you can get them. Have you tried MORF? They send free binders to trans masculine people in the UK - all you have to do is pay postage, which is a couple of quid. Or if you get pocket money or birthday/Xmas money, you can save it up & buy one for yourself. You could also try to run errands, or do odd jobs in your neighbourhood to earn a bit of cash. Failing that, once you're 16 you can move out & claim benefits - and you can buy a binder with those. So you
will get one, even if not right now.
In the meantime, there are other things you can do. You describe yourself as chubby: well, if that's bothering you, you can always eat more healthily by cutting out any junk food and snacks. You can do more exercise free of charge by walking around your neighbourhood, or going up & down stairs at home or in shopping centres. You can do some basic weight training by filling shopping bags with tins of food & lifting them. You can do resistance training at home in your bedroom using nothing but your body weight. Not only will this help you feel better about yourself, but it should also reduce your need for antidepressants... and it will give you better results should you choose to have top surgery.
The best thing to do is to pick one goal at a time and put all your energies into working towards it. If you're autistic, that's a very natural thing to do. So why not pick something like the Sandyford, or a specific goal weight, and do everything in your power to get yourself psyched and ready for your first appointment there?
And try to ignore your mum & her girlfriend as much as possible. They're probably going to say some horrible TERFy things over the coming months; if they do, try not to let them get to you. Look up the stages of grief, and you'll see your mother go through them all because although she's not losing you, she's grieving for the loss of her idea of having a daughter - a loss that she's probably feeling more acutely because of her radfem ideas. It's early days and it can take our families quite a long time to come round, but they usually do. If they get abusive (including emotionally or mentally abusive), contact ChildLine - they'll help you up to age 19 and you can contact them online if you struggle with phones. We're also here to help if you need us.
See? It's not hopeless. You
can get there, and you
will get there. You're just in a temporary rut at the moment, but that will change. My life has improved enormously since I started transitioning, and my only regret is that I wasn't able to do it sooner.
Also, congrats on not self-harming for a month.