Yes, you can.
Your NI number has absolutely nothing to do with getting NHS treatment - it is used to work out how much tax you pay on your income... it's like a SSN. An NI number is only issued to UK citizens aged 16 or over who have reached working age, or to people who have moved to the UK and wish to work.
When you register with an NHS GP, you will be automatically issued with an NHS number (this is usually issued to UK citizens at birth/when they have their first medical appointment as a baby)...which is a unique number in the NHS system, used to track your medical history. However, you will need proof of your UK address (and your passport and BRP) in order to register, so you'll likely have to wait until you have some mail in your name.
If you need treatment quickly, you could probably either register as a temporary patient until you have proof of address, or you might be able to get treatment at a Family Planning Clinic, but it may depend what kind of treatment you need. You can also use the A&E for emergency treatment if you have to.