Coming at this from an alternative angle but I really, really wouldn't bollock her or punish her. She's fifteen years old and her Father has encouraged her to drink. Be that one of five, he encouraged it. It tastes nice - rum does - and she's had one and clearly thought 'Oh this is fine!'
She has no idea about tolerance or limitations, alcohol is a very precarious balance and he needs to understand how harmful it can be. Give her appropriate consequences about taking the rum but seriously, seriously educate her on how to look after herself when drinking.
Let's be honest, who here wasn't going out at a weekend at 15/16 and getting drunk with their mates? Very few of us. My Mum was very strict with alcohol and I knew I couldn't go home. The few times I did she would shout, take my phone off me, refuse to let me have friends around so I just didn't go home. I was a 'messy drunk' and was into my early twenties. Why? Because no one taught me the signs to look out for, no one taught me how to pace myself etc.
She'll feel as sick as a dog today, leave her be. Express how naive she's been and say you'll talk about this when she's fully sobered up. Then say how disappointed you are for her stealing the alcohol and that she really, really scared you. Take coke and then a show her how strong a drink should be, then explain to her how many drinks are too many.
Talk to her about the strengths etc and reiterate WHOLEHEARTEDLY that no matter what situation she is in, what she has does, who she is with or where she is that she cannot ring you to come and collect her and bring her home. Don't stop her from going to London - what would that achieve? Treat her like an adult and see how you get on. She may learn from this.