If it's really mobile BB, as the forum in which you posted, you want to transfer to a new place then you just take the mobile router with you.
You sure you don't mean fixed Home BB? If so, you need to follow EE's Home Move process.
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To phone EE CS: Dial Freephone +44 800 079 8586 - Option 1 for Mobile Phone & Mobile Broadband or Option 2 for Home Broadband & Home Phone
ISPs: 1999: Freeserve 48K Dial-Up > 2005: Wanadoo 1 Meg BB > 2007: Orange 2 Meg BB > 2008: Orange 8 Meg LLU > 2010: Orange 16 Meg LLU > 2011: Orange 20 Meg WBC > 2014: EE 20 Meg WBC > 2020: EE 40 Meg FTTC > 2022:EE 80 Meg FTTC SoGEA > 2025 EE 150 Meg FTTP
