Who does what and when really depends on the nature of the emergency. There is no simple answer. In the case of a RTO there is an automatic handover of control. Hence you may well change from being PF to PNF as you are rolling down the runway. Once the Capt has brought the aircraft to a halt and asked what is wrong you would be directed to carry out ecam actions. The capt would make a PA requesting the passengers to remain in their seats whilst he also has a chat to the purser and assesses the situation. He might be trying to get information concerning the emergency from the purser or calling the tower to ask if they can see anything. It might necessitate waiting for fire crews to arrive on scene to make an inspection.
Taking the RTO example once the standard calls have been made and the aircraft is decelerating and it looks like it will safely stop on the runway you as PNF (your the f/O) would make a quick call to ATC. All you need to say is "Tower xyz715 stopping on the runway, standby". The standby is so that you don't get into a longwinded conversation with them. As helpful as ATC are in this situation your sole priority is to safely stop that aircraft on the tarmac in front of you.
The checklist is there as a backup to the memory items that the capt will have undertaken and you follow through with it to make sure that he hasn't missed anything eg: left an engine running for the passengers to be sucked into. In our RTO example Line 2 of the checklist is there to give more information to ATC in another radio call. You might for example want to advise ATC that you have had a fire warning and request emergency services. They know you are stopped because hopefully they can see out the tower window (nice clear VMC day) that you have plus you made an initial call telling them so. If it is decided by the captain that you are going to evacuate then he will have briefed you of his intention to evacuate and will ask that you advise ATC. Simply say "Tower, xyz715 evacuating present position request all available emergency services". Then get out of the aircraft using any available exit as per SOP. Of course VHF1 will only be available so long as the batteries are left on.
Exactly when you advise ATC of your need to evacuate will depend on the nature of the emergency. And who makes the call depends on who is PNF or PF at the time which will be also be impacted by events such as auto handover of control.