Hey Mia, yes, I've been flying in diapers pretty regularly, both on short and long haul flights, and I have always been changing in the airplane toilets no problem. I usually put my spares and wipes in an inconspicuous black bag like the one you put shoes in and grab that from my backpack in the overhead locker before heading to the lavatory. It's not the roomiest place to perform a change but it works for me, the upside being that it breaks the monotony of the flight, and it saves me time at the airport. As joemama mentioned, it can be tricky to dispose of the used diaper in the lavatory trash bin, if it's not badly soiled it's easier to slip it in there without rolling it up.
As for security checks, I never went great lengths to avoid being found out, as wearing diapers is not illegal in any way. Full body scanners like you have in the U.S. aren't that revealing in my experience, I went through them once with a pretty wet diaper (I think it was a boosted 24/7, so nothing lightweight) and the only thing the scanner highlighted was the sweat spot on the back of my t-shirt from carrying a backpack. The only time I was given a patdown was in Berlin, when I went through the metal detector wearing my plastic pants with metal snaps (which I avoided since then on). The security agent definitely felt my diaper but not a word was said and no weird looks were had, as that was probably far from being the first time he inspected somebody wearing a diaper. Bottom line, even if they have to pat you down or request you to show them your padding, you bet it is going to be an awkward moment for them too, and a tricky one at that because they're basically asking you to expose your disability, and if they did anything in a less than professional, courteous and polite manner it would take very little for you to sue them and get them fired.