'Wrong road' means a movement in the wrong direction, which this wasn't; it was a diversion via a different branch. This happens fairly often for a number of reasons on the Northern line, the most common of which is late running as the journey via the CHX branch is quicker than via Bank.
The on-train DMI is not directly linked to the signalling system and requires the driver's input to change, unlike the platform display which work directly off of the signalling data.
For the driver to say that s/he was diverted with no warning after leaving Kennington isn't technically true. Whilst s/he could have been diverted into Kennington 1 only after leaving the previous station, as soon as a diversion was inputted by the signaller an audible alarm would sound in the cab and a message would appear on the train operators display stating 'Destination Change' - the destination on the TOD would also change from, for example EDG-Y (Y = Bank) to EDG-X.