Was bored earlier and decided to sacrifice an Extreme Green Potion for the sake of science, and it turns out that Mille Aiguilles' actual attacking speed, and not just the start up delay, is affected by Extreme Green Potions and Speed Infusion. I assume it would also be affected by Advanced Yellow Aura, Zero's Aura, Inner Ability, etc, but I only have SI and potions currently.
I recorded a minute or so clip holding down the attack key with just booster, booster + green potion, and booster + green potion + speed infusion active and counted 18 'hits' of Mille Aiguilles (18 stabs, starting from the very first stab and ending on the first frame of the 18th) and checked the times for each speed:
1.467 seconds w/ just booster
1.233 seconds w/ booster + green pot
1.166 seconds w/ booster + speed infusion + green pot
Those times aren't counting the start up delay either, I originally thought green potions would only affect the start up delay but it seems they work for the speed of the stabs too. Would be interesting to see if this works for other 'hurricane' type skills that have actual character animations (Ring of Ishtar does I believe? can't think of others though) to see what works.
I can upload the video later if necessary but if you're curious and have fraps and a video editing program you can do it yourself pretty easily to confirm. I'll try Asura later since that has attacking animations too.
Apparently it does affect the speed of Asura too. Counted 20 movements (excluding my irritating Star Cape which seems to move all the time for some reason), with no speed buffs and with decent SI + green potion + booster:
0.966 seconds with no buffs
0.7 seconds with booster + extreme green potion + decent speed infusion