I could be wrong, but I think they might have started dating pretty much right away while filming the pilot in March because by June, they were very SHOWY in terms of their feelings.
Especially during the events.
I was watching the Special Features, and Kat talked about how her and Elyes were basically stuck in the car filming that airplane/car scene for 2 full days just the two of them. So I think that bought them together. At another point in the behind the scenes stuff, they were shown to be play fighting....
So I'd say at least by April, they were dating...I'd assume. Could be sooner!