I can tell you with almost 100% certainty: no banks in the UK use any 2FA other than SMS-based.
I spent December last year looking for a new bank to move to. One of my criteria (not the most important but it was on the list) was better-than-SMS 2FA.
No one offers it. There may be some niche, loosely-based finance org that does but none of the banks or Building Societies do.
So, unfortunately, you need it in the UK.
Barclays, with standard current accounts, provides several methods none of which are SMS. There's a separate pin-code device (called Pinsentry) that does TOTP and challenge-response, or passcodes for both telephone and Internet banking.
Yeah, Nationwide has the little PIN code device which definitely helps with transactions but not logging in: that's still old-school.
Most VoIP services support SMS. Still no cell phone hardware required.