Monzo Bank no longer requires any proof of address. They are doing this specifically to address the issue immigrants have with access to banking [0]. They are a fully licensed and regulated UK bank, and have built their entire infrastructure from the ground up (zero legacy mainframe software) [1]

You can mail your debit card to any address you choose, no verification required.

“When you sign up through the Monzo app, we’ll ask you to take a picture of that ID and a short video of yourself. That’s it! You’ll need a UK address where we can send your debit card, and then you’re good to go.”

[0] https://monzo.com/blog/2018/05/29/getting-bank-account-immig... [1] https://monzo.com/blog/2016/09/19/building-a-modern-bank-bac...