Upvote:4
They will allow you to use the visa in your old passport if it is the one issued immediately prior to your new passport. They will usually mark your new passport with "VIPP" (Visa In Prior Passport).
You donβt have to transfer your visa - you can carry both your old and new passports when travelling to or from the UK instead.
Source: https://www.gov.uk/transfer-visa
You did not provide the amount of unspent leave on your visa, but if it is longer than 90 days, they may require you to get a transfer of conditions during your next visit. This is a very straight-forward process which does not include the assessment and decision-making that happens during a visa application.
Finally, regarding your change of address, if you want to play it strictly by the book you can prepare a change of circumstances notice and present it to the Immigration Officer at your landing interview when you arrive.