I'd read and re-read the NATS site several times before doing my application, always reading 15 days between stages.
I had my stage 1 on Monday in Southampton. Echoing the comments of the last few pages - if you read the ATC pack in the invitation email and learn it you give yourself a good chance to get through the first test. There were about 80 in my sitting, and just under half went through to the cubes and diagramming.
For the second part I got the impression that they don't expect you to finish, but it is possible - just do as much as you can as accurately as possible. I managed to complete about 95% of the questions on both tests and felt they both went well.
We were told by the lady leading the course that they were trying to turn around results faster and would hope to get them done within 7 days.
I had my tests on Monday, and got an email Tuesday afternoon inviting me through to stage 2. Now booked in for the 16th which was the first available date when I clicked - seems places are filling up from the messages above.