No, you don't actually have to do anything. You can just carry both your old and your new passports with you when you travel.
So, at UK immigration, you show your new, valid passport alongside your old passport with the valid visa in it (I do this with my US visitor visa - the visa is valid until 2021, but my passport expired in 2013, so I just hand them two passports at US immigration).
Alternatively, you can apply to transfer your visa into a BRP (you would have to send your old and new passports to Durham), but it costs £107, takes several weeks, and it's not mandatory.
See:
https://www.gov.uk/transfer-visa
