Although they all can be used interchangeably, the best way I would describe it is:
Consistent means happening with no change in variation, for example someone always coming to work at 7 in the morning.
Constant means that does not stop, as in time. For example someone keeps tapping their pencil through a class.
Continuous, the best way to put is that it never stops. Such as a river, it will stay flowing without stop.
Hope this helps!