I didn't think I'd have a problem with approaching girls I'd never met before, cos I think of myself as a pretty social guy. So on the way to meeting some friends on Saturday, I decided to try a cold approach before I met up with them (which I haven't tried before).

I was wrong, I found it really difficult to make myself stop any girls to start a conversation with them. I decided I wouldn't let myself go and meet my friends until I'd attempted to start a conversation with at least one person, I ended up walking about for over an hour before I finally managed to start a conversation with somebody and even then it was just a couple of guys that I did speak to. But by this point I was getting pretty fed up with my inability to start a conversation, so just wanted to speak to anyone so I could let me myself go and meet my friends.

How common is this level of "approach anxiety"? (it feels wrong calling it that, seeing as it also happens when talking to guys I don't know).

I'm not expecting that fixing this aspect of my life will be easy, I'm just trying to get an idea of long it'll take to become competent. Even if it takes 5 years, that time is going to pass anyway so I might as well be a better person by the time I get there. I was hoping more for something in the 6-month range. But after seeing how difficult it was for me on Saturday, I think 6-months might be a bit optimistic (I only started looking at this aspect of my life a couple of weeks ago, so I'm very new to all of this).

That level of "approach anxiety" is very common, especially if it's was your first time.
But if your serius about getting it handled you could probably get rid of it in 6 months.
Start slow. Make 1 approach a day, and than make more approaches as you get more confidance.
Don't think too much about what you're going to say. Speak your mind.
Just my 2 cents..
Hope it helps