Damn, bringing out the challenge round, huh
@RedNeck?
First off, lesson #1: if you're planning to approach later than a few minutes from now, or wait for some event you're going to have to monitor her to wait for it to occur, it's risky. Can still work, but introduces a lot more wiggle room for things to go off the rails.
I don't know if you were next to her (sounds like not), but if you were and the eye contact was very good you could always slip her a note... something silly and cute, ideally: "I know you are in a meeting but your eyes are adorable and also that's a really cool set of laptop decals

-RedNeck"
As for going after her after creepy dude... all right, well here is what happened, she solicited an approach, but you didn't approach, and instead this guy did. So she is going to be pissed at him, but also pissed at you. It happens a lot with the "she signaled you, you didn't go in, then some weirdo did" pattern.
I have run after girls in a handful of instances like this and my experience has been that if you REALLY like the girl it still works; the rest of the time you get blown out. You have to have a forceful enough, cheerful enough frame to overcome her disgust at the guy and annoyance at you leaving her in that position by not approaching then + now, after you put her through all that, being Johnny Come Lately.
So if I do not really like the girl, I would just let it go.
However, if you do really like her, then you just run after her, and open her ASAP from the side (not behind) and go, "Hey, sorry, I meant to talk to you back there, just didn't want to interrupt you, it looked important. Man, that Starbucks is going downhill. I'm gonna have to change to Costa Coffee. I hear the beverages there are excellent and it's totally weirdo-free. It's probably like the biggest selling point for Costa... 'no weird people allowed', I think it's actually part of their slogan," etc., where you are just pacing and leading her to positive emotions, just straight running your mouth, not asking her questions, not talking specifics about the creepy weirdo guy that will bring her mood down, just trying to elevate her mood back up, with humor, etc.
Even then, it is a tough battle, because she is all shaken up and weirded out, and you are trying to shift her out of that before asking her to decide on having a conversation with you. If you can't run your mouth long enough to lead her out of that emotion you're toast.
Even if she starts asking you a few questions, you need to be on high alert for whether they are just polite questions, which they probably will be, so you can redirect them and keep pumping the mood up and chilling her out.
But yeah, this is super low odds, it's a really tough turnaround.
Thus why only worth it if you're really into the girl.
You can NOT express interest in her, can NOT ask her questions about herself, can NOT tell her anything meaningful about yourself, can NOT invite her to do anything until she is totally out of that shaken mood. That's the real challenge... can you pull her out of that before she excuses herself and bails? Probably not.
If you can run your mouth long enough to calm her down, then get past the polite questions into an actual real, chilled out conversation, then you can salvage it. But getting there is very tough.
Worthy of a shot though if you're up for a challenge and the girl is cute enough!
Chase