It looks like you are expected to apply as soon as you start a job:
From :
http://www.inlandrevenue.gov.uk/nic/ynino.htm#4When Do I Apply For A NI Number?
No one has a legal right to a NI number but there are circumstances you are legally obliged to formally apply for one and to register for NI purposes.
Criteria for applying for a NI number
If you do not already have a NI number you must apply for one as soon as you start work or you or your partner claims benefit.
You must be over 16 years old and resident in Great Britain.
If you satisfy all the above conditions except for being resident in GB, and you still want to apply for a number, you must be liable to pay NI contributions or want to pay voluntary contributions and would benefit from doing so.
Providing you satisfy these conditions, you should contact your nearest Social Security office (This is a link to the DWP (formerly DSS) website) and ask for an appointment to be interviewed for a NI number.
At the interview you will need to be able to prove your identity.
You can find out information about the types of documents you should provide to help establish your identity in leaflet GL25 available from any Social Security office (This is a link to the DWP (formerly DSS) website).