They would not fall, but flying speed would be reduced to 0
The Fly spell reads:
The target gains a flying speed of 60 feet for the Duration.
Since flying speed is a speed, This means that when speed is reduced to 0, flying speed is also reduced to 0. The rules on flying speeds being reduced to 0 read:
If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.
The flying rules say that when flying speed is reduced to 0 the flying creature falls. However, since these rules essentially say that if the flying status is created by magic, with the fly spell as an example, in this case the character would not fall, so they would simply be unable to move after using the cunning action during this turn.