Hi, thanks for all the answers.
Unfortunately, I did call the DPW instance today and she answered me very short that I need a 3 month residence and an address in the UK in order to apply for an NI number. I tried to explain my situation but for her the 3 months residence was the only guideline. That was it. Then I called HMRC and they passed me back to the DPW, as you realise, going in circles.
I'm a bit worried, since my pay slip actually shows that I contributed as required but at the end I won't be able to claim it when needed.
The only thing I want is being able to get the famous E106 document to proove my local health care department (in my own EEC country) that I actually conributed so I won't fall outside the system.