Registering for a mobile phone :
Denmark
Registering for a mobile phone
These sites may help you decide which carrier is best for you and how the system works. You will need both proof of residency and your CPR to get a mobile phone contract.
Easy Expat - Mobile Phone
As for your UK mobile. Our guess is that you will be returning to the UK from time to time and may want a low cost way for people to contact you when you are there. So why not cancel your contract, but switch to a "Pay as you go"? This is a number that most of your friends/family will already have, so record your voicemail explaining that this phone is now only on when you are in the UK, and tell them how to contact you when you're not. Be careful when choosing your PAYG plan as some expect you to use your phone every few weeks to keep it active.