You don't need an NI# to get registered with a doctor. I've been here for nearly two years, been registered few times with different docs in different areas and only just now got my NI# since I just started working. Once you get registered with a doc, they'll give you an NHS number, which is different from your NI#
You need to contact Jobcentreplus, call them or go in. Me (US, in UK for two years) and a flatmate (EU citizen, in UK for five years) called them on the same day and got different but essentially the same procedures. They took my address and mailed me an application. I filled it out--just asking my visa details, address details, etc. I mailed it with a prepaid envelope they sent me. My flatmate got an interview about a week later and filled out the application with the officer there--she just brought her passport and lease agreement.
It was really easy and surprisingly quick, got my NI# letter in about ten days. The actual card hasn't come in yet, but that apparently takes six weeks. However, the letter has your NI# so you should be fine to work once you got it. I have another flat mate who started working before her NI# was issued so it depends on your employer. My employer apparently wanted my NI# before adding me to the payroll so it may differ in your case.