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Whenever you're planning a journey, consider if you really have to fly. I am Asperger myself, and I avoid flights (although not for the exact same reason as you).

Between The Netherlands and the United Kingdom, there is absolutely no need to fly. There are three ferry links from The Netherlands to England: one to the south, one to the centre, and one to the north. Taking the ferry as a passenger only is a very peaceful experience, as the terminals were built in an era of far more passenger travellers, and most travellers nowadays drive onto the ferry by car. The times I've taken overnight ferries in Europe, passenger terminals were deserted to the level that I thought I must be at the wrong place.

If I stay on the same continent, I travel by train whenever possible, even if it means 3 days on the train. Northern Sweden to Southern Spain, Toronto for example, from northern Sweden to southern Spain, or from Toronto, ON to Jasper, AB in Canada). It's quite possible, if you have the money you can book fully private compartments on many sleeper trains. There's a wealth of information from The Man in Seat61.

Note that there are some trains with airline-style luggage checks. On the Eurostar, your luggage has to be scanned. On many Spanish trains too, although you might be able to hold on to your plush. People with more (recent) experience on Spanish trains may be able to comment on that.

Now, suppose that you're travelling to a place where travelling without flying is almost impossible (Australia, South America, sub-Saharan Africa) or extremely time-consuming from Europe (east-Asia, North America). In this case, I have little to add to Kate's answer. It may get easier if you do it more frequently.

Edit: StrongBads answer reminded me of the Autipas. It's a special card you can show to airport personnel (it was reportedly inspired by an incident with an aspie flightspotter on Schiphol Airport). Personally I wouldn't like such a card, but perhaps you would.