If a fuse in one of the three phases breaks before reaching the VT (Voltage Transformer) used for measuring the generator terminal voltage, and this situation is left unattended for a long time, what issues could arise?

 

If you are running, you should trip a protection relay, probably Phase imbalance or whatever gets there first.
If you are offline, you will either trip once the exciter starts or once the generator breaker closes on a similar protection relay, it depends on which VT the fuse has blown.
If you do not trip in either of these scenarios and you find this issue, time to pull out the drawings. You have a bigger problem.