All I know is you'd be eligible to reclaim overpaid tax. For example, an individual works for the first six months of the tax year (April-Mar) and then leaves the country. Their wages would have been taxed weekly/monthly on the basis that they were going to be paying for the full tax year. As they have left half way through the year, they most likely have overpaid.
There is a form P85 which is to be completed to reclaim and P45 from last job is needed. the non-residents FAQs page from HMRC.
For non-resident tax questions contact: +44 151 210 2222
For non-UK resident Nation Insurance questions contact +44 191 203 7010
You wouldn't get NI contributions paid back I don't think. You may get NI entitlements transferred. For instance, when I first arrived, I was entitled to claim Jobseekers' Allowance as I was from the UK. Payment was delayed until they had confirmation of my NI position from the UK.
It took me 3 months to get my tax back, but most of the delay was because I had no P45 as my former employer hadn't issued me with it upon leaving.
HTH,
Rich
(I'm not qualified to give advice, but have generally found the HMRC helpful if you ask them for an answer to a particular tax situation)