Sounds like you aren't using a punting phone otherwise you would just change your SIM card 
Do punters really need a SIM card at all?
Most SPs are happy to use WhatApp, and many prefer it.
And there is still regular email, of course.
You need an active internet connection, but free Wi-Fi is everywhere.
And as far as central London is concerned, it's not very sensible to be standing in the street with your phone, using your mobile data. Much better to get off the street and dive into a coffee shop or some such, and use their Wi-Fi...
It's easy to forget that a smartphone is a mini computer and internet enabled device. A SIM just allows you to make calls and send/receive SMS...
Without a SIM you can still use mobile banking apps to transfer funds. And you can still use map apps.
But you still need a phone number to set many things up to begin with, but you can use any number.