1.) Career
Employer - Your employer needs to know of a change of address so they can update your contact details, but more importantly they need to know for your payroll.
Inland Revenue - One of the most important places that needs to know when you change address is Inland Revenue. They'll need to know for your taxes and you can find out how to update this information on Gov.UK.
National Insurance - Your National Insurance (NI) number is frequently used by the government to help identify you and is required by the DVLA, HMRC and your employer. That means it's extremely important to update your national insurance information. You can find the phone number and opening times online, but you'll need your NI number ready to fully change your information.