Your National Insurance number is a lifelong identifier that never changes, regardless of how many times you change your name. What does need to be updated is the name linked to that number in HMRC's records and, separately, the DWP's records. Keeping these records accurate matters for employment checks, benefit eligibility, state pension calculations, and other purposes that reference your NI number.
If you are employed, your employer will update payroll records when you provide your deed poll, and the change flows to HMRC via the Real Time Information (RTI) system. You do not typically need to contact HMRC directly. If you are self-employed or file a Self Assessment tax return, you should update your name via your Government Gateway account or by calling HMRC. The DWP maintains its own record of your NI number, which may need to be updated separately if you claim benefits or are approaching state pension age.
You do not receive a new National Insurance card when you change your name — the NI number itself is unchanged. Some people ask whether they need a new NI letter confirming the number under their new name; HMRC can issue a confirmation letter on request, but in most situations your deed poll is all you need to update any organisation that references your NI number.