I've recently moved home and these are the steps I had to take to change my address with my current bank and Monzo.
My bank: I look on their website which says I can phone up and do it over the phone. I phone up, go through the automated options and then I'm told that actually I have to go into a branch with ID and proof of address because apparently I'm not registered for 'telephone banking' (why?!).
The nearest branch to me is open 9-4.30pm on weekdays and 9-1pm on Saturdays. I generally work at these times so I have to take time out of work to go into a branch, queue for 10 minutes and eventually get it changed.
Monzo: I open a live chat within their app, answer a couple of security questions and the address is changed right away. It took about 5 minutes at most and I could do it while working on other things.
My bank also charge a small fortune for foreign transactions and contactless won't work abroad. Monzo is free and contactless has so far worked fine in 3 different countries.
The UK banking industry is massively behind the times with regards to customer service, facilities and technology. The potential for Monzo is huge if they can pull off what they're trying to achieve.