I met this lady a few days ago and we had a nice conversation that lead to us exchanging numbers. For the past few days I have been trying to get her out on a date, but she insists she is in London for one dance rehearsal or another. I think she is really into me because she's always texting and asking me questions about myself and is seemingly apologetic at the fact that her unavailability it out of control.

My situation is this,

I've got my friends birthday on a Saturday and I initially had wanted to see her instead after work, but she's unavailable due to her dance class. I can't do Sunday unfortunately because I have got to be in London and I got an interview on Monday that I want to spend Sunday preparing for. Not to mention, I got a short that I am shooting.

I am keen not to convey the idea that I am available so I don't want to book a date on Sunday (Plus I got other things to do). I had mentioned to her that I will not be available for the following two weeks (no idea why) so I am thinking I might be able to squeeze her in next Sunday, but I am also worried that so much time would have passed that she may eventually lose interest.

Whats the best course of action at this point? one that doesn't make me seem needy and one that does not take too long.

Get her on a date ASAP. Your right - if time passes nothing will happen.

I wound up going to my friends birthday instead, and so far we've been texting (actively keeping delays between each text). I'm conscious not to prolong the date, but I'm also conscious of not making her my number one priority. Instead, I am looking to expand my options. I have a girl who works in a bar with me that I am tapping up for availability. Assuming that doesn't go anywhere, I also have another girl who might be available for some other extra curricular activities.
I think the right thing to do now is to keep cultivating other options so that I am not so dependent on this one girl. Stay tuned for updates.