Use your EE sim on the scheduled relocation date. Expect that there will be no service signal from both of the SIM (your old and new). Wait until 9 PM to check if the process is successful. See Step 4 on monitoring the progress.
How to Move From Three to Giffgaff
This mobile provider is never limited in changing mobile number even from the Three network.
If you are a former Three user, here are the steps to transfer number to Giffgaff from Three successfully: