It is incredible that there is so much lack of training around in employers and their personnel departments. So many of them still think that on being given details of solely a person's National Insurance number ("NINO") they are protected against prosecution for employing an illegal worker. Merely having a proper (not temporary) NINO used to protect the employer from such a prosecution ... but no longer.
Download this PDF :-
Changes to the law on preventing illegal working: short guidance for United Kingdom employers
Printed copies of that booklet were sent to all employers about 15 months ago. OK it is 20 pages long but if you are able to print out a copy and take it with you to interviews, together with at least one List 1 item, or two List 2 items, then if the potential employer still refuses to employ you solely because you do not currently have a NINO, well as it says in the booklet, they might be guilty of facial discrimination.
Good employers are now asking
all successful job applicants to produce one List 1 item, or two List 2 items, and are recording that information, in order to take advantage of the statutory defence as mentioned in the booklet.
If you don't want to print out the whole booklet, I would suggest print out at least the pages listing the List 1 and List 2 items.
Good luck with finding work.