Not only was this my first ever cold approach but it was one that can be rather challenging too, the feared parent and daughter set.

here is what happened:

i was sitting in a cafe near st pauls in central London, i saw this stunning HB 9.5 girl walk in with her mother, i felt i had to take action as i am 22 now and feel that i really should take action now and always approach every girl i see that is attractive unless she is on her own with a man her age. anyway i walk up to the mum and daughter, i say to the mom how attractive her daughter is, they both smile and thank me, i introduce myself to the girl and things are generally positive as both her and the mum are smiling albeit in a very surprised fashion. i talk to the mum too and ask them what they are up to, they say they are heading to some seaside town in england, i feel myself getting very nervous having not done such a thing before, as i can tell the mum is eager to leave and so i straight up ask the girl if she would like to go out with me, she smiles and laughs a bit, just seems a bit shocked Lol, her mum says she lives in scotland which is about 400 miles away, i say that is no problem but the mum insists that would be too much of a commute for me, things are getting uncomfortable as the girl has not said anything, i ask if she has facebook and she just says "no i dont" i should have asked for her number but things were so tense with her mum and everything i just let them go.

how does this sound for a first cold approach?

Good job approaching them like that. That takes balls.

I would have told the daughter she was beautiful directly, and then included the mom in the conversation. If you're going to tell a girl she's beautiful, tell it to her.
I would have followed up with saying that you were working on improving yourself and you were going to talk to the next beautiful girl you see and all of a sudden she walks in in slow-motion with her hair flowing in the wind *They laugh* (The reason I would say this is because it would make the interaction seem more like fate - and girls love chance reactions in falls into their romantic view of how they are supposed to meet a man - and it makes you seem like an honest guy)

Once they laugh, they will probably relax and be a bit more receptive. Just focus on rapport for a bit - they gave you a rapport topic by talking about the seaside town - "Oh it's beautiful there do you guys go down there a lot?" "That place is a great spot I highly reccommend it".
Once you've made them comfortable you can get a number close. Just something like "I know you're probably not in town for long, but I would love to get to know you more, if that's okay with your mom (acknowledge the mom). Maybe we can go out for a bite to eat later?"
You exchange numbers.

Girls don't like feeling uncomfortable in a situation - they assume if you're making them tense now, all your interactions will be tense. You have to be comfortable and make them feel like spending time with you would be a comfortable, exciting, fun thing.

I think you did everything well except it was just probably a little tense since it was your first cold apporach, but that will go away with time. Also a little rapport to make them feel connected to you.
And knock it up into sexual attraction by saying you would like to get her number. Don't do anything waaay sexual with her mom around.

That's my personal take on it. I haven't done cold approaches for a long-ass time now, since I've been with my girlfriend for 4 years now.

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Thank you burnskee a really great answer, you were absolutely right that i should have built more rapport around them and done stuff like asking them about their holiday etc, that said i could really tell the mum just wanted to get away fast which made things very tricky, plus the fact that i was talking to a girl that overwhelmed me with her beauty(i will get used to that after more approaches!). i still feel that i should have gone for the number after i asked for the girls facebook which she did not have but i think my mind went blank at their lack of interest in me at the end and the fact that i just pulled off such an approach which was a shame as when i approached them their reactions were in essence positive.