The speed increase will stack because the effect does not have a duration
The Dungeon Master’s Guide errata (pdf link) added the “Combining Game Effects” section which states (emphasis mine):
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. (…) Game features include spells, class features, feats, racial traits, monster abilities, and magic items. (…)
Since game features include class features one would think that we can conclude that two instances of Blade Flourish’s speed increase would not stack; but in this case, there is no overlapping duration. We can conclude this by looking to the Essentials Kit, which includes a “Effects with the Same Name Don’t Stack” section (page 3) stating (emphasis mine):
(…) But when two or more effects have the same proper name, only one of them applies while the durations of the effects overlap (a duration is a time span of 1 round or more). (…)
This defines what counts as a duration and in this case, the Blade Flourish’s speed increase lasts until the end of the current turn (less than 1 round), thus you can benefit from the speed increase multiple times if you took multiple Attack actions.