For a NI number, go here:https://www.gov.uk/apply-national-insurance-number
For a NHS number, you simply need to register with a local GP. You'll likely not "get" your NHS number as it's not issued as a card. But as soon as you register with your GP, you'll be in the system.
Bite the bullet and apply for your provisional license. It can take awhile to get your UK license, so the sooner you start the process, the better.