Sorry, but I don't understand this as I had no trouble at all getting my NIN card decades ago. But things have changed, I guess. They want to make it impossible for any foreigner to survive here. Brexit. Heard of it? The shameful, official vote that says we are not wanted. The vote to boot us all out, if they can anyway. I mean, the anti foreigner sentiment was always there, always has been, but the british public made it official two years ago, bless them. This can be a very racist country. Saying that, it's not the UK who have built that monstrosity of a wall, is it. As a California, I am deeply ashamed and embarrassed by it. But we are off topic.
If I were you, I'd kick up a huge fuss and don't let up until you get what you are entitled to, if you are entitled to it of course. but no, you will not be able to work without a national insurance number. nobody will hire you. But I have found here that if you keep on things, and people, eventually you get what you want. Not always, though. HMRC and DWP and other government bodies can be the most stubborn trolls you've ever come across and they may not back down. But it just seems to me that if you are naturalised, then you should get a NIN. Find a way. Find a way to meet the criteria they impose.
I honestly dont' know why anyone would want to come here now. Honestly. The people voted against having anyone foreign living here. So for anyone who hasn't yet moved here, seriously think about that. It is not easy to live somewhere you know you aren't wanted. You never, ever, feel fully accepted here. Ever. Remember that before you come. x