Connection: Belfast to Amsterdam to Florence

You will go through immigration at Amsterdam as that is where you enter the Schengen. Forty-five minutes is tight but Schipol is an efficient airport.

Brexit (whenever or if ever) is irrelevant.

I like Schiphol. Signage is excellent, so it will be easy to find your gate, but it can be a long walk (though doable). As noted above, you will go through immigration/passport control. Schiphol has made a lot of nice improvements in its passport control process over the last few years, so it is more efficient than it used to be. There will also either be a lane for tight connections or a Schiphol employee calling out for people with tight connections to speed you through.

Historically, you have not had to go through security when transferring at Schiphol. The last time I flew through the airport, though (4 weeks ago), there was a random security screening immediately following passport control for those headed OUT OF the Schengen Area. That's the opposite direction you will be going. It was just a lady randomly choosing people who passed her to go through a security screening. I was chosen. The screen consisted of me getting x-rayed and my bag getting x-rayed, but it was quick (virtually no line). For that trip, I flew into Europe through CDG/Paris, so I don't know if there is anything like that in the direction you will be going (INTO the Schengen Area). I have rarely been stopped going in that direction to be sniffed by a bomb-sniffing dog, but that takes seconds.

If you are on a single ticket issued by KLM, you are golden. Even if you miss your flight, KLM has 4-5 flights per day to Florence, so the airline hopefully will get you on the next one (happened to me one trip going to Berlin because my incoming flight was late). Just go to a kiosk and scan your boarding pass; your new boarding pass for you new flight will print out.

Dave, Thanks for posting such a thorough description of we can expect when we arrive in Schipol. We are going to make every effort to be on our 8:35 p.m. flight to Florence because it is the last flight of the day. The "worst case" scenario would be to miss our flight and then have to take a flight the next morning, or whenever. Fortunately, our time in Florence is our own, so if we arrive a day late, we wouldn't miss anything.
Maybe we'll try to alert one of the flight attendants of our tight connection, so s/he can give us specific info/gates.

Pay attention to signage, you want immigration for connecting flights or "in transit".

Edit: I guess to expand a bit, perhaps not on a flight from Belfast, but typically once landing, the biggest crowd all heads to Immigration as Amsterdam being their final destination. You do not want that, you would be put out beyond the secure zone and have to go through security again.

You want to look for the connecting flights immigration, tends to be shorter, and except for screenings, no additional Security delay.

I am happy to report that we made our connecting flight: our plane arrived at Schipol on time from Belfast, walked rapidly toward our gate at the far end of the airport, passed through immigration quickly, and arrived at our gate just as the passengers were boarding.
Two Happy travelllers!