I can confirm what people above said. All I needed was a NI number. And a bank account.
With the NI number you don't really need any bills etc., but I had trouble obtaining a current bank account (even though I had a student one for the past 3 years), as they kept saying I need 3 years of residence in UK (after requesting to speak with the branch manager and asking her if they told rich Russian oligarchs the same reason and not giving them a bank account they accepted my uni years as residence). However when my gf came over as well, they got her a bank account no questions asked, all I did was phone the council and told them there are now 2 adults in the house and to put both of us on the tax bill. With that piece of paper and me with her at the appointment at the bank, they didn't hesitate at all.