Yes! You will receive the payment from your employer even if you do not have the NI number. In such a case, you have to pay a high amount of tax as you will not get the code that you are under tax-free threshold. This code has to be processed by your company by using an emergency tax code till the time HMRC issue you an NI number.
You will have the option to work for your entire tenure without the NI number as there is no legal requirement to have the number. Your employers may ask you to provide the NI number in the shortest possible time. This is happening because many payroll systems have been made in such a way that NI number has become mandatory.