The actual rule doesn't care what it is that you're carrying... just whether the total of what you're carrying on a train/tube fits within a 1m x 1m x 1m space.
A full-sized bike in bits probably doesn't, and you probably can't move it around easily.
Would a bicycle considered a bicycle once disassembled? the ticket guard seemed to be happy with it.
Guess we're at the mercy of the ticket guard.