Legally, you can take the money.
Morally, I don't think you should.
You need to have an open and honest conversation with her about this, laying out the options. Of course, she's going to be upset. All her life, she's been led to believe that she has this £30k waiting for her, and right when she's an age where she may stand to benefit from it, you're telling her you're going to take half (at least).
Her being upset doesn't make her an entitled brat like some posters have said. Anyone (including them but they won't admit it) would be upset in this situation. You've dangled a carrot in front of her face for years and when it comes time for her to have a bite, you've cut it in half.
That doesn't mean that there isn't a way for you to take the money you need for the house-and maintain a relationship with your DD, but you need to lay out the options clearly and concisely for her.
The house is X price, and it's the only house suitable within touching distance of your budget. All the other houses are unsuitable or in areas that even your DD doesn't want. (She's 19-why is she getting a say?)
If you are to afford this house, you need X deposit and you can't make that deposit up without that £15k. If she is to give you (and make it a choice for her to give it and not for you to take it) the £15k, you can get the house and you will put her name on the deeds/make sure she has a share of interest in the property. This will benefit her in the future.
At 19, she's not looking at this with long-term thinking. She's seeing that money that she's been told about for years being taken away from her. That is bound to cause upset.
You need to navigate this with understanding about that, and not just "Well, it's really my money and though I promised it to you, circumstances have changed and I'm breaking my promise and taking it back."
This isn't an easy, cut and dry, situation, and neither of you is entirely wrong here. It's a matter of how you handle this and making her feel like she has the option. Are all the other house options utterly terrible? I'd also agree with other posters that, she's 19, she shouldn't have a say in where you move to. She won't be living there that long, most likely.