for the purpose of this question EEA = EU+EEA+CH
Can anyone explain how we will all be processed - is it standard that everybody goes through the same channels? Or are we all going to be separated. I saw on BBC Breakfast this morning (20 July) that at Rome airport it seems British passports do finger prints and passports at one e gate after another.
You are going to be separated. There are provisions for non-EEA family members of EEA citizens to use the EU Citizen lanes if travelling along with the EEA citizen, but they don't apply to unrelated persons.
In the non-EEA citizen lanes, you'll either go to a automated kiosk, eGate or manned booth (depending on airports, if EES is enabled at all). Then answer questions about stay, take prints and photo...
If first step is a manned booth, likely that are everything will be handled in one swoop. If first step is a gate or kiosk, you'll be sent either to another gate or a manned booth and then be sent on your way.
Is it likely that we will all be separated with different queueing times?
Yes, I don't know about Verona, absolutely plan for much longer queues than your EU travel companions.
What are people’s experiences so far at foreign airports especially travelling with a Schengen area passport.
EU Citizen lines/eGates are much faster IME, but it is random at the end of the day. It'll depend on passenger load, airport layout, walking distance, how many staff the border authority decided to send that day... You can roughly estimate flight load by looking at the arrivals' board for flights arriving around yours.
In my experience with one of the worst airport for this, CDG, as a French citizen, it can take from 5min to 30min, estimated non-EEA lines were > 60min at both times, but it's CDG so it's bad all the time.