@MipMipMip
I think a lot about children who feel they're trans these days, because I'd like to have children at one point and I try to imagine being in that kind of situation and it's absolutely heartbreaking, even without having kids. I can't imagine the pain a parent must go through to witness their kid being so upset, and to know the risks and potential regret. I know what it is like to be on the other side though, so I know one thing for sure, that restricting/forbidding your kid from doing the things they want to do will not end up well.
I hated my name all my childhood and preferred being called nicknames, I dont thinkt here's anything wrong with that, as long as the name is not a dead giveaway (if a Susan wants to be called a John for example). I would try and encourage my child to do "gender neutral" things instead of things associated with the opposite gender, to take it slow and experiment "safely". Encourage them to not take anything at face value online, to be critical. Like it or not, if they think they're trans, they'll eventually end up on a trans forum or other online community and they'll see a lot of happy post-transition people who only brag about their good results. Seeing a lot of happy posts like that can give them a false view of what post-transition looks like, but nobody talks about the occasional Male to Female transgender who still looks like a dude in a dress after 10 years of hormones, or that the suicide rates pre- and post-op trans people remains almost the same.
Ignoring it completely will not make it go away and will potentially make it worse. I think the most important thing is to build trust with your child by showing them that you accept them and support them no matter what - don't pretend you're happy about it, do tell them that it's difficult for you, but show support and tell them that their happiness comes first. If they see that you are willing to "sacrifice" your own discomfort for their happiness, they will appreciate that (I know this sounds stupid, of course you want them to be happy, but as a kid who's been in that situation, it meant the world for me that my mother told me this and we grew much closer in our relationship afterwards).
Talk to them lots. Try to get them to talk about what makes them sad, angry, happy. Understanding what "triggers" their disphoric feelings can help you suggest them much less drastic solutions. I think talking to them yourself will get you much further than seeking third party help, unless it's a very good therapist, which you probably wont find on the NHS, and private ones are usually very pricey. Me being able to talk to my mum openly and very honestly about these stuff helped a lot. Hating your body so much that you want to become someone else (even if you dont consciously realise this) is a terrible feeling, remember that they feel this way and that they are very desperate for any solution that promises to ease this pain.
Tying back to the point I made earlier about restricting/forbidding stuff: my dad was like that and I soon started viewing him as an "enemy". "He just doesnt understand, he doesnt know what he's talking about". I lived with my mother, but I know if she was like that, I would have attempted getting on with transitioning even without her support. I've seen a ton of forum posts online on some trans sites where a youngster as young as 14 posts about how their parents dont support them and then a lot of trans people chime in, saying they should absolutely buy hormones behind their parent's backs. No matter what you do, your child will eventually be an adult and they'll do it without your support. You will only alienate them if you go down this route, unfortunately.
Unfortunately I dont have much in terms of resources for parents to read, I know two websites that deal with this kind of thing, one of them just asked me for an interview and they both have some good resources probably:
4thwavenow.com
www.transgendertrend.com
Hope that helps!