I carefully looked at screenshots to put it together--this is immediately after the Cyber Planner has taken the Queen which the Doctor sacrificed on g6. There's no mate in three here but the Doctor actually did win the game with his move.
The Doctor can now move knight to e7, forking the King and Queen. The Cyber Planner can only respond by moving his King to h8--Doctor takes Cyber Planner's Queen, again putting Cyber Planner into check. If the Cyber Planner takes the knight with the pawn on f7, then the Doctor can use his f1 rook to take the Planner's rook at f8, indeed a mate in three, but Planner doesn't have to take the knight, he can move the King back to g8 instead to get out of check. At this point, the Doctor can put his knight back on e7, the King has to go to h8, then the Doctor sacrifices a rook on h7. Once the King takes it, the Doctor delivers check by moving his other rook to h1. It's practically mate because the knight covers g6 and g8. The Cyber Planner can only delay the inevitable by sacrificing a rook at h4. But of course, that's mate in six, not mate in three.
I thought it was a draw at first before someone else in my chess club pointed out the six move mate. Cybermen are confirmed to be overconfident.
I mostly relied on this screenshot to get the positions;