Harry Allens
New EE customers can transfer or port their old phone numbers to use the new service. Also, they can port it when moving from EE to another network of their choice. Switching networks does require a new SIM and normally all the needed instructions are printed on the package or answered in the EE online FAQ.
If things are unclear or if you find yourself in a special situation (e.g. having terminated the contract with the previous provider), then your case will require assistance. Simply call the EE number for the customer services and you will be helped by an EE representative.