Although it's been a few years now, when I arrived in 2007 I called the number, and they gave me an appointment at a local Job Centre where I did the rest of the application.
I got a number within a couple of weeks or so without any hassle, so I'd say just call the number on your first link. There's no need to use a service.
You can work without a NI number. Just tell them you've applied for it and you'll give it to them when you get it. When I was in that situation, nobody seemed to be too worried about it.