Eberron and the forgotten realms are not separate planes, they are actually separate crystal spheres. Crystal spheres are basically just different solar systems that dont necessarily have the same planes or exact physics.
The concept of traveling between these crystal spheres is called spelljamming. It's sort of like space but also like the multiverse it's very strange but it's how dnd handles campaign setting jumping.
If you want to put a war forged in the forgotten realms it would probably be easier to just have an npc invent them somewhere in your world, otherwise you'd have to weightwait until the party is kind of high level or at least accompanied by a high level arcane spellcaster, and then reinvent the spelljammer because they haven't been used in thousands of years in lore.