Your allowances for earning interest before you have to pay tax on it include:
your Personal Allowance
starting rate for savings
Personal Savings Allowance
Please see the guidance at:
https://www.gov.uk/apply-tax-free-interest-on-savings
If you earn interest over your allowance but under £10,000 and are employed or receive a pension, HMRC will change your tax code so you pay the tax due automatically.
The code for the current year will be based on how much interest you had in the previous year.
You need to register for Self-Assessment if your income from savings and investments is over £10,000 in the financial year.
You can find further information here:
https://www.gov.uk/check-if-you-need-tax-return
Tax on savings interest: How much tax you pay - GOV.UK (www.gov.uk)