Staff speciality starts at the moment he is signed.
It is important to note that the games arent happening in real time. When you are watching the game live, the GE has allready run the sim and is now showing you the results. If your player gets hurt in the first quarter, you cant buy a doc with the 'taping injuries' specialty and expect his specialty to effect the game, as it has allready been simmed just prior to you being able to watch it.