From the main thread:
" * Using your phone’s email client to access your email - OR - your phone’s browser to access your webmail via browser (using the same email used initially to sign up with Monzo - SEE BELOW**), look for the very latest email from Monzo with a Magic Link and then follow the instructions in that email (it will have a box to tap on). Users have reported that Magic Link emails are not received, usually when using icloud email accounts. This is because the Magic Link emails are interpreted as Junk. Search all icloud email Junk folders using both the mobile device email client AND via browser to find the latest Magic Link email. Please note it will be from ‘…@monzomail.com’ and so it is worth marking this domain as ‘not junk’ (whitelist it)"
Source: Monzo App sign-up/login/access issues - common things to try