The lungs stay in the donor's body until the recipients hospital is ready to take them. Recipient hospitals are called when the person is going to be taken off of life support. The organs cannot be procured though until there is brain death determined and once they are taken off of life support they have to have cardiac death. A person has to have cardiac death (heart needs to stop beating) within an hour of being taken off of life support or else nothing can be procured (taken).
I was told at 12 in the afternoon that organs were available/there was a match and my transplant didn't start until 9PM. The person was kept on life support until my doctors could get there to get the organs and he/she was brain and cardiac dead. I think that my docs got the organs around 8PM or so...maybe just a little earlier.