This is an interesting situation.
You ask :
What would be the most polite way to approach this situation? [and get your recording].
There is a primary principle here Respect the other person and give them control over their image, and voice. Their Image and Voice are after all a part of who they are, their self.
In your specific situation I would say that if you are going to record her you need to do the following in order to still have a chance of being polite.
1) don't interrupt until they are done working (filming here news piece).
2) As soon as they are done (or take a break) ask her permission before sending off your video to friends.
Personally I don't think you should film other people with out their permission unless you are trying to protect yourself or another victim from a crime or corruption.
When my oldest daughter was 2 years old, my wife and I had taken her to a park. While there a man, who was already taking photographs with a friend, appeared to start taking pictures of my daughter. My wife was furious, and I was bit uncomfortable about it as well. We didn't know why he was taking pictures of her, I would like to assume the best of intentions for him, but as a parent in today's culture you become a little paranoid.
I've also been in situations were some one was holding up their phone as if they were recording me. I didn't like that either.
Basically what I'm saying is, unless it is a friend of family member who you are close to and know won't mind or if you feel the need to record evidence of a crime or of corruption, you should probably ask the persons permission first. It's just polite.