Aaron and Celina speak before starting their shift. Later Celina, Bradford and Chen take the John Doe to the hospital. In a later scene they arrive at the hospital and ask Luna about him saying he was brought in the previous day. When Aaron speaks to Celina at the station she refers to the conversation they had "this morning". However it was already established it was the previous day.
No matter how they paid for it, Lopez & Harper would never be given permission to stop what ever they're doing and fly to a different country (they never specify what country John and Bailey are in) and participate in an unofficial investigation that has nothing to do with any cases in Los Angeles.
Just because he's a patrol cop in LA this does not give John any special privileges in whatever country he happens to be in. He could be cited for impersonating a cop and tampering with evidence by local law enforcement. No matter how "practically non-existent" they may be.
It is mentioned that the walk in refrigerator is working on a backup generator but that clearly is not true as the fans for the evaporator are not spinning.
Lopez and Harper arrive on the island, yet Nolan and Nune couldn't leave the island because there was no ferry.
However, when Harper and Lopez arrive, a motorboat leaving the island can be seen in the background. Presumably the detectives arrived at the main island and found a ride, but it would be much more complicated the other way around.
When Chen and Bradford are talking in the hospital, an extra walks by in green scrubs and very briefly looks directly at the camera and smiles.