NI number is national insurance number, it’s the closest to an ID number there is in the UK.
From the .gov.uk page
Who uses your National Insurance number
These organisations need to know what your number is:
- HM Revenue and Customs (HMRC)
- your employer
- the Department for Work and Pensions (which includes Jobcentre Plus and the Pension, Disability and Carers Service), if you claim state benefits, or in Northern Ireland the Department for Social Development
- your local council, if you claim Housing Benefit, or the Northern Ireland Housing Executive
- Electoral Registration Officers (to check your identity when you register to vote)
- the Student Loan Company, if you apply for a student loan
- your pension provider if you have a personal or stakeholder pension
- your Individual Savings Account (ISA) provider, if you open an ISA
- authorised financial service providers who help you buy and sell investments like shares, bonds and derivatives - you can check if your provider is authorised
https://www.gov.uk/national-insurance/your-national-insurance-number
Your NHS number isn’t important and isn’t even really needed to get medical treatment, they’ll normally just ask name and DOB. I don’t think I’ve been asked for my NHS number in the last 10 years (or ever).
If you don’t have an NHS number the system will just generate one for you first time you register with a GP, but you’ll need to apply for an NI number on arrival if you don’t have one.