First, as others have said, wear a quality, thick diaper. Goodnights won't cut it. The goal is to be able to go the entire flight without changing, because trying to change in an airplane lavatory would suck. I personally am an ABU fan, so any of their products should work just fine. That being said, still bring an extra diaper and some wipes in your carry-on just in case.
Second, unless you have TSA PreCheck, put your diaper on AFTER going through security! Your diaper will show up on the big body scanners they use, and you will be questioned about it. This happened to me once. I managed to avoid a pat-down by just telling the TSA agent straight up, "I am wearing a diaper." (As an off-topic side note, if you fly regularly, TSA PreCheck is a really good investment. Not only do you get to use the old metal detector (which doesn't detect diapers), you also get to skip the line and keep your shoes on.)
Third, make sure you stay hydrated. Not just because, as Tetra mentioned, the dry air on a plane will dehydrate you, but also because it will make your pee smell less, which is courteous to your fellow travelers. Needless to say, do not soil yourself on a plane. It's not fair to make everyone else smell that.
Fourth, just relax. People generally don't notice anything beyond their own limited sphere of existence, and they certainly won't suspect that you're wearing diapers. Just act normally, and no one will ever know.
Finally, since you mentioned that you're going to London: London-Heathrow Terminal 5 has a changing room separate from the bathrooms with a changing table easily big enough for an adult. I had a layover in Heathrow a few months ago, and I was able to change my diaper in there.
I hope this helps!