Today, frustrated by lack of usefulness of number #2, I tried going back to the Job Centre, taking with me details of the second-closest one as well. The Job Centre was just as closed as ever - but this time, a helpful guy in a car told me to keep going down the street I was on. And sure enough, further down the street was a fully open, entirely functional Job Centre Plus. I went in - and the man at the welcome desk gave me a sheet of paper with another number to call on it - and this time, there was even a partial explanation for why the number I'd been trying for the past two weeks wasn't working. As of today, the overworked Stoke Newington call center was closed and replaced by something better.
Back at home this afternoon I called - and to my amazement, the call was connected, a helpful person took all my details, and then agreed with me that twenty minutes was too little time to run across town to Tooting with all my uncollected documentation for the mandatory NI interview. Unfortunately, since I couldn't do twenty minutes notice, I have to settle for five weeks notice. But I have an interview date, right at the beginning of August! Once I finally, finally had the right and then updated information, it was all very easy.