If it is a single phase, 120/240 volt generator, then any current leaving on Phase A will return on Phase B, and vice versa. Any 120 volt loads that are running at the time will split the 240 volts in proportion with their resistances, since they will be effectively in series. That will mean that some might have a voltage well under 120, and others might have a voltage well over 120, either one of which can cause equipment failures.
If it is a three phase, 120/208 volt generator (not related to your question, but I'll mention it just for completeness), then the situation is similar. Current leaving the generator on one phase will return on the other two, and any 120V loads will see out-of-normal voltage levels.
Finally, regardless of whether the generator is single phase or three phase, if the only loads that are on line at the time the neutral breaks are single phase, and all on the same phase, there will be no current, since there will be no complete path for current to follow.