After Lady Melisandre lit the Dothraki's arakhs, they waited few minutes before they charged. Why did it take so much time? And especially, who told them to to storm because Jon Snow and Daenerys were up on a cliff and did not send a signal?
Presumably Jorah or Qhono told the Dothraki to start the charge as Jorah was leading the Dothraki battle group but Qhono was essentially their "leader".
Why did it take so long? Well they had a battle plan, which they didn't exactly follow, but it probably said something along the lines of start the attack at X time or when you see X. And to be honest it didn't exactly take that long really, a couple of minutes is nothing in the scale of a full battle.
I don't have any canon source to back this up, but it looked to me like they just decided to attack on their own. Their line was a bit ragged, they seemed to leave in individual clumps rather than all together. Also, this is very much in character for a Dothraki horde. These aren't any sort of highly disciplined knights, they're a horde and one that's used to destroying anyone and anything in their path.
So, I don't think anyone gave the order at all. I think they just couldn't be bothered to wait any longer and, ignoring the plan and their commanders, rushed headlong to their deaths.
No one told the Dothraki to attack. They are a horde and they just took off on their own. You notice this if you watch closely, since they take off and almost trample Jorah as he tries to navigate his horse around them. Plus he initially looks confused, then gets a look of "I guess we're doing this."
For all those complaining about Dany, she sees them die (as shown by her expression and exchange with Jon) then she takes off out of anger. Jon follows and this chain started by the Dothraki is what messes up their battle plans.