Joffrey takes a drink from his chalice, and drops dead while eating pie. There are a few reasons why I think the Tyrells/LF used a combination of poisons, so that Joffrey would need to consume both parts (the wine and the pie) in order to die.

1. In Cressen's POV, it seems like after he drinks the strangler, he is struggling to speak immediately after. In his condition, it seems unlikely he would be able to swallow, much less eat multiple bites.
2. Tyrion makes himself look extremely guilty by pouring out the wine in the cup. However, its reasonable to say that neither the Tyrell's nor Littlefinger could have planned for this to happen. In this case, if the wine had remained, food testers wouldn't die from drinking just the wine. Its better to have people question whether Joffrey was poisoned at all rather than question who did it.
3. Margaery drinks from the chalice before the poison is added. While its possible that Olenna and LF communicated a signal to her, she's sitting right next to Joffrey. The risks of either poisoning her, of Joffrey noticing the signal, or of her attracting suspicion by not drinking are low on their own, but together there's a lot of moving parts which create opportunities for error.

This leads me to think that part of the poison was in the pie, and part of the poison was in the wine. After drinking the wine, part of the poison is already in Joffrey's system, but its not until he eats the pie that he receives the second dose and it is activated (and then he dies) .