You'll need to tell us so that we can update our records, and you may need to apply for a grant of probate. This will mean that you can legally handle your partner's finances for accounts where you're not a joint account holder. The government has a website that can help you: www.gov.uk/applying-for-probate
The estate of your partner may need to pay inheritance tax to HMRC. This will be due six months after their death, and the executor will need to estimate the value of the estate and contact HMRC to find out if inheritance tax is owed. If they leave their inheritance to you, inheritance tax won't be charged if you're the spouse or civil partner of the deceased.