HI Nippos,
Welcome to Apple Support Communities. We understand that you're having an issue with accessing a verification code on your iPhone. We'd like to help.
From what you've described, it sounds like this is your only trusted device. Is that correct? Is it possible that you have another Apple device or computer that you're signed into with the same Apple ID? The reason we ask is that you will need to verify that you should be accessing this account before being able to view the data in the iCloud website.
Check to see if this resource helps at all: Get a verification code and sign in with two-factor authentication - Apple Support
If you still have issues, you may need to start Account Recovery or replace/repair iPhone to gain access again. How to use account recovery when you can’t reset your Apple ID password - Apple Support
Hope that helps.
Take care!