The time on the phone is right but the one time password won't work.
The time alone is not enough. As said, you should also do in Authenticator->Settings->Time correction for codes.
Also I've noticed in the past 5 years that (rarely) the websites can have incorrect time sync and that sometimes gets "handled" by trying to log in either in the first seconds after the minute, either the last seconds, either just after the minute has passed.
Does anyone know of a way to get the keys out?
If you manage to actually use the 2FA codes and the number of them is not too big (under 20), then it may be less effort to actually change (turn off+turn on) the 2FA on each website and this time make sure you save the seed too somewhere, just in case. (Also you can use something smarter than Authenticator).