I am allergic to dairy, and have soy milk instead. Is it something that's easy to buy in Western Europe, or should I look into bringing some with me?
The countries we are visiting are: England, France, Germany, Switzerland, and Italy.
In all the countries you have listed, soy milk is easy enough to find; most supermarkets will sell it.
In France, you should be able to find it in Super U, Auchan, and Carrefour, and maybe in some Aldi and Lidl too if they're big enough. In Switzerland, I got many links to Reform Haus; you can find those in most of the northern part of the country. In Germany, most (if not all) supermarkets will have it, even smaller ones. Same in Italy; you'll find it with the boxed milk section, or the refrigerated aisle, along with other soy products. And in the UK, you shouldn't have to worry, even small Co-operative or Spar stores will have it sometimes.
And if you ever can't find it in those, you can try Asian stores. Those are a pretty safe bet.
Basically: Yes, soy milk is pretty easy to find in Western Europe.
Yes, I've seen it (and various other non-diary milks) in markets (even smallish ones) in the UK, the Netherlands, and France — coincidentally, because I don't actually drink soy milk. So I'm thinking if it's so common that I remember seeing it even though I don't look for it, it must be fairly common.
It may be useful to take a list with you of "soy milk" in the native languages of the countries you're going to visit, as sometimes it may not be where you expect to find it in the market, so if you need to ask someone where it is, it'd be useful to know what to call it. The level of English spoken in Europe is very, very good, but uncommon terms like "soy milk" may be a reach for someone who doesn't speak English well.
In Germany and Austria you can buy soy milk in most supermarkets. If you are interested in a broader selection of soy and tofu products you will find a Reformhaus in every larger city.
These supermarkets are specialized on food with special ingredients.
In Germany, even discounters like Aldi have soy milk and some other soy products. However, if you want more soy products than just soy milk, it's probably better to go to a supermarket like Rewe, Edeka or Kaufland.
The supermarkets provide other plant based milk (e.g. milk from rice, oats or almonds), too.
This is the survival vegan tip I learnt on my travels: whenever you arrive to a country where normal supermarkets don't have soy milk (which is rare, most of them do), go to an Asian supermarket or try to find an asian neighbourhood (chinese, vietnamese, etc) and you'll get those things there. Sometimes soy milk will necessarily have sugar (ough!) like in many places in South America, but you can always find it there.