In a recently published FAQ by Mojang, the developers have confirmed that Minecraft Dungeons will not include cross-play at launch. And, while cross-play between the Switch and the PS4 version was already questions, from the look of this answer it seems like Xbox One and Windows 10 will not have cross-play at launch either.
Mojang has the following to say in the FAQ:
At launch you´ll be able to play online multiplayer with players on the same platform. Our goal is to let everyone play together regardless of platform, and we’re looking forward to enabling cross-platform play in a future free game update.
In the FAQ Mojang has also confirmed the existence of a beta. It is only for PC players and very limited in scope. It’s a technical beta for a small sample of the content. It’s aimed on testing the online gameplay.
The complete FAQ can be found here.