Taxiing back to the other end is a "backtrack". At a controlled airport like that, the backtrack would need a clearance.
When you land at London City, you will roll out and turn off onto that little turnaround bay, where you are "clear of the active" and you would then need a backtrack clearance to return to the other end.
It would depend on the traffic level. If you landed and someone was right behind you, you might need to wait for the other a/c to roll past and then you'd get a backtrack clearance to pull back out onto the runway to taxi to the other end. If it's not busy, the tower controller may give you a backtrack clearance as part of your landing clearance so you can land, turn around and proceed straight back. The controller can't give a landing clearance to anyone else until you are clear.
In any case, the main thing is that you can't proceed back without permission from somebody.
Note that London City is now controlled offsite using special cameras and sensors, but it works the same way.