As you are on a spousal visa, your NI number was not generated for you and put on your BRP (I think that only happens for some work visas). Instead, you will need to apply for a NI number through the Department for Work and Pensions (DWP). To apply, you'll need proof of address, your BRP and passport, and your partner's NI number.
The application process involves attending an appointment at a Jobs Centre Plus.
Booking that appointment can sometimes be tricky. You probably won't have a problem because you're planning to start work soon, so just tell them that and they'll book you in. (I wasn't job-seeking when I got mine, so they didn't want to bother with giving me an appointment so I could apply.) At the appointment, they'll verify your right to work and your identity by glancing briefly at your BRP, then they'll fill out a form on the computer that you easily could have done yourself. You'll wait a few weeks, and your number will appear in the post.
You can start working before your number arrives. You'll be taxed at the 'emergency' rate, but you will get a refund for any excess tax withholding from HMRC once your record is updated with your NI number.
You probably already have this information, but in case you don't, the number to call to apply is
here.
