I worked part-time for three years over here while at uni on a temporary NI based on my d.o.b. When I decided to stay on and start working full time I applied for a proper one, even though as an EU citizen I technically didn't need one to keep working. I did just walk into the the local job centre, they took details. It took three months for an appointment to come through, then I went in for an interview with stacks of documents in tow, and a week later my shiny NI card showed up in the post. I think the amount of paperwork was slightly complicated by the fact that at the time I didn't have a permanent address but was couch-hopping, and as well as convincing them I wasn't really, really homeless
(with some help from my uni welfare staff vouching for me), and that in Sweden you don't carry your birth certificate with you but a statement from the population register (which had to be translated into English)... and so on.

But mostly painfree.

:o)