Pisa Airport

Why do you think there will be immigration from one EU country to another? Both are signatories to the Schengen treaty, so no immigration at either airport. You shouldn't have a problem with the transfer. There is a special train called the Pisa Mover from Pisa Central to the airport train station and it takes just 5 minutes. At that time of day, they run every 9 minutes - schedule here http://pisa-mover.com/en/shuttle-timetable/. You can also watch a traveler who has recorded the entire transfer on Youtube if you need reassurance. Whether the baggage can be checked through will depend on your airlines.

The real issue I see is that this looks like a 2 ticket booking. ?

If so, the retrieval and re-checking of the bags impacts the amount of time you have between flights. And adds having to go through security.

Another thing to factor in is that the plane may not park at the terminal building, in which case there is the added time of taking buses to/from the aircraft at its remote parking location

Be sure you have all this factored into your transfer time.

As noted above, flying Pisa to Athens, via Rome, is akin to flying from Chicago to Orlando , via Dallas. Customs/immigration is a non-issue. Its the transfer time in Rome that could be the problem.

Thanks for your advice - we are novice travellers. Rome is not the issue as we have a 3 hr stopover. I am more concerned about allowing enough time once at Pisa. Would really like to avoid the earlier train.

You should be at the airport by 9:30 with no problem. Pisa airport handles only 5 million passengers a year, so not too busy.

Many Thanks.