When a player has Droll & Lock Bird in his hand and the opponent creates a chain, in which there are multiple drawing effects (like Reckless Greed, Jar of Greed, Legacy of Yata-Garasu or Accumulated Fortune) then the Droll & Lock Bird owner needs to wait for the whole chain to resolve before he has a chance to drop Droll & Lock Bird. That of course mean that his opponent will draw all cards that he is to draw from the chain he created and after that Droll & Lock Bird will make it so that he cannot draw anymore.
Reason is because when the chain is built, the opponent hasn't drawn any cards yet, and since Droll & Lock Bird has to be activated when an effect that adds/draws finishes resolving, this is clearly not the time for it yet.
When the chain is resolving, Droll & Lock Bird still cannot be activated since effects cannot be activated in the middle of a resolving chain.
Once the chain has finished resolving, now it's the time for Droll & Lock Bird to be activated.
Reason is because when the chain is built, the opponent hasn't drawn any cards yet, and since Droll & Lock Bird has to be activated when an effect that adds/draws finishes resolving, this is clearly not the time for it yet.
When the chain is resolving, Droll & Lock Bird still cannot be activated since effects cannot be activated in the middle of a resolving chain.
Once the chain has finished resolving, now it's the time for Droll & Lock Bird to be activated.