No connections will be open and is open until the late afternoon (i.e. until last TATL flights leave)
You will pass the connections desk, then passport control (as you are entering Ireland/UK CTA) and then there is an optional security screening and into the terminal, no customs check as that will be done at FCO
If you have arrived from a 'safe' country e.g. US/Canada/EU then no screening is required.
However once a flight from an 'unsafe' country lands then security screening will become active as they cannot separate out arrivals.
The late morning Emirates arrival tends to be the trigger