Ok, as someone who runs a payroll for the family firm, let me clear this up once and for all...
Its a myth that its up to the employer to apply for a NI number. This is only if you have a NI number and you have LOST it, not if you require one.
Firstly, you start work, and you tell your employer you will apply for a NI number. In a meantime, tell the employer to use a nil NI number on payroll (this to conform to Inland Revenue rules). The employer cannot use a temporay number.
Next, apply for a NI number interview at local DSS. They will have a slot within 4 weeks.
At the interview at the DSS, bring your first payslips, and if issued, a contract of employment from your employer or some kind of letter confirming employment.
Within 3 to 4 weeks, a NI number will be sent to your home address.
Of course, the above info only applies if you have a visa allowing you to work in UK!
The current threshold for NI is around £88, so in theory you can earn under that amount and never pay NI, but still it is a requirement that all employees must have a NI number.