From the original script, it's not immediately clear. On the one hand, Johnston ("R.J.") is clearly fluent in Mandarin:
RJ stares at the faces which begin to peer in through the car windows.
A grinning TWELVE YEAR-OLD slaps a poster against the glass.
R.J.(points at the ideogram on the poster) "China awake..."
and at the same time, the Emperor shows a strong understanding of written and spoken English:
Pu Yi points at a portrait on the cover of the Post — a special
"Independence Day" issue.
PU YI: Who is this "George Washington"?
Since they can both communicate in each other's languages (and since the Emperor is eager to learn about the outside world) I'd imagine that they would speak English whenever possible.
Elsewhere in the Forbidden City, Mandarin would obviously be the lingua franca for general conversation.