They will be fine, they really will! If mine was -- and he is a sensitive little soul! -- yours will be too.
My cat is a shy and retiring type, sensitive, clingy and very attached to me, so I had worried that the entire thing, with the separation, all the strangers, the long flight, everything, was going to permanently scar his psyche! He was freaked out from start to finish, but after a day or two being settled into his new home, he was bouncing right back to his normal self. He was extra affectionate with me, he did snuggle even more, but also started enjoying exploring his new digs, after a day or two. It helped to have brought his old familiar toys, bed, and other things. And even though he is a shy type, he very quickly became fascinated by the new flat and all the different things here to explore. Within 24 hours he was playing and running about as if he owned the place.
Don't worry if it takes a few days -- besides thinking "What the hell was that?" about the journey in general, they are quite literally jet lagged too, just like humans! I'm pretty sure my nervous nelly of a cat probably did not sleep the entire ten hour flight, and yours may not have either. Right after collecting him he looked so freaked by the entire experience that I was seriously worried -- but if he made the recovery very quickly back to his happy little self, I know that yours will too, don't worry. I really feel that even though this kind of journey is a lot for them to cope with while it's happening, it is not a lasting trauma once they see their human again at the other end, and he is exhibiting such relaxed happiness that I have never regretted putting him through that one day of freak-out.