I've been doing this for a week now, and it seems to be going pretty well. Here is my sleep chart for the first week:
The first couple of days went surprisingly well: I woke up about half an hour later than I would normally wake up with the alarm. The time I woke up was slipping back eachday though, and on the fourth day I woke up at 6:30am which made me think maybe it wouldn't work. After that it went back to before 6:00am and has continued to be in the 5am - 6am range most days, which is when I want to get up.
The other thing that I was worried about was that I would sleep in, then not be tired at nap time and then my whole sleeping pattern would get messed up. I'm happy to say that this hasn't napping, and my naps have generally been about 1.5 hours in length which is better than before.
Overall I'm finding the experience of waking up without an alarm to be a lot more pleasant, I'm feeling less tired in the day and generally enjoying this. I'll write another update in a week's time.