I have traveled by checking ATX builds in the past, but now recognize how unbelievably reckless that was. Unless you're willing and able to fully insure your luggage, I would highly recommend either carrying on a mini ITX or shipping a larger rig ahead via a reputable carrier.
If you decide to go ITX, there are tons of options. @JokerProductions has just shared his RVZ02 build for LS, which will be great. This is kind of a mid-high range ITX build.
https://forum.teksyndicate.com/t/lan-syndicate-mini-itx-travel-build-in-a-bag/96131?u=shoring_fragments
Logan also built a fairly rugged system a couple of years ago. You'd have to update the internals a bit, but something like this could also work. Beware that the PC-TU100 doesn't have great thermals, so you wouldn't want to jam a high-TDP card in there.
https://teksyndicate.com/videos/colugo-our-new-itx-gamingediting-rig-build-video
You could go higher-end all the way up to 5960X + R9 295X2 or lower-end all the way down to something like an APU-only system and still have a good time. It's really up to you and what kind of performance you want, and on what budget. If you can narrow it down a bit, we might be able to give you more concrete suggestions.
I'm not familiar with UK carry-on luggage restrictions, but in my experience traveling through Europe, anything sub-20L that isn't too bizarrely shaped will probably fit in an overhead compartment. Silverstone, Cooler Master, Lian Li, Fractal Design, and Antec all have suitable options, and then there is the whole world of boutique ITX cases. Something larger, like a BitFenix Prodigy, is probably not a good idea.
To echo @VXAce above, air cooling is likely to cause you fewer problems at security, and it's good to detach both your CPU cooler and GPU from the motherboard before the flight, and to add some cushioning if possible.