In a private sale between two FOID Card holding Illinois residents, the three day waiting period for a handgun and the 24 hour waiting period for a long gun theoretically starts when the the agreement for purchase/sale is actually made, with the physical transfer of the firearm held up until the waiting period is complete.
In my experience with every firearm I purchased or received through an FFL transferring dealer, the clock starts ticking on the relevant period when the dealer has a completed ATF Form 4473 in hand and actually calls the IL State Police for the approval number, regardless of when a prior agreement to purchase was made and/or the money changed hands. Then it ultimately comes down to when the FFL called in for approval.
That is my experience. What happens when the FFL receives a 'delay' or there are multiple handgun purchases within a five day period is another story.