Why does Britain use "Way Out" rather than "Exit"?

At public transport interchanges throughout the English-speaking world (and where there are English signs for the benefit of travellers in non-English-speaking countries), the exits are marked, appropriately enough, Exit. The one exception seems to be Britain, where they're marked Way out. I'm just back from Northern Ireland, where Exit is used.

Does anyone know the reason for this?

Way out and exit mean approximately the same thing. Is it too simple an answer just to postulate that our authorities may have happened to standardize on different terminology because it sounded better to them, or because it sounded more naturally-spoken in the respective country? In other words: there isn't a 'reason'. You're overanalyzing.

I think it is to differentiate between the normal everyday way out and the fire exits that should only be used in emergencies.

Because "exit" is a Latin word (meaning "he/she/it goes out"). Why use Latin when a perfectly good phrase based on Anglo-Saxon English serves the same purpose?