The notion of movement speed being per round and not turn is not very useful.
In general, the stuff a character is able to do during a round is limited to their turn, the major exception here is reactions, which we'll get back to. Your speed is a metric which informs how far you can move on your turn for free. In addition, all creatures can take the Dash action to move equal to their speed on top of that. Some creatures, like Rogues and Monks, can even take the Dash action as a bonus action on top of that again, allowing them to have moved three times their speed on their turn.
Effects which force movement come in two primary categories,† either they force the creature to move on their own turn using their normal movement, eg. fear. Or, they force the creature to use their reaction (which is usable on any turn) to move, eg. dissonant whispers. In the latter case, the movement is in addition to the normal movement, just like when taking the Dash action (other than which turn it happens on).
†: There's arguably also the category of spells and effects which push or throw the creature, but it should be fairly clear that a creature’s speed statistic has no effect on how far it can be thrown. There's also the case of infestation which makes the creature move a set direction and distance (5 ft.) which can be considered pushing conditional on the creature being able to move (ie. not grappled, paralysed, etc.), but its use of movement.