The only real impact was that I was locked out of anything that required two-factor authentication.
Old phone stays in a safe, and only pull it out to add new TOTP codes.
The email providers I use all require 2FA, but as long as I had paper backups of recovery codes for those I could probably use those to get back into most other accounts. No comment on whether I do have aforementioned paper backups, or whether I'm just relying on having a couple of devices with access to the codes, and then hopes, prayers and obsolete e-waste in drawers...
Of course, travelling makes the puzzle even harder, and that's when it's often even more important to not get locked out of accounts. I had intended to offer a suggestion to worry less, but instead I'm going to go away and think about this more myself.