Doing so will be quite the task. Inevitably, however, it will help out AOROX. Sadly, I can not tell you on my own. Contrary to this roadblock, however, there are still ways. On the other hand, you'll need to look deeper than just the surface. Rarely do I do this, but it needs to be done. Daily do such dilemmas cross my mind.
"Revenge...is like a rolling stone, which, when a man hath forced up a hill, will return upon him with a greater violence, and break those bones whose sinews gave it motion."- Jeremy Taylor
Do you have an IRD, or TFN?
Unfortunately, even if you were born in the UK, it's difficult to get your NI number, if you weren't present when they are usually issued.
Things that will work as proof of identity, for this, are limited.. for a list, it would be best to check for your Birth Certificate (or valid coppy), any card you may have for IRD/TFN - not sure if they still have those in physical form, these days, or your passport (though you're getting a passport, and that should work, as you seem to have found out, already - your current one may work, but could cause delays).
Take as many as you can, as having more than one strengthens your identity.
Given you didn't get your passport stamped, it's going to be really hard for you to provide proof of valid entrance into the country.. The official at Heathrow really should have known that, and issued you with a page (unless you got a stamp, but not an indefinite stay, one).
You'll need records of employment, as well, so make sure you have those to hand (records of payment will usually work for this, at a pinch).
Documentation of past travel may be difficult to obtain, if you've had more than one passport, in the past, as well.