The fact that the key is rotating but not moving the vertical rod suggests that the problem is with the lock tumbler itself, but it could also be caused by a problem with the linkage between the tumbler and the rod, or with the latch mechanism itself.

To determine the exact cause of the problem, you may need to remove the door panel to access the lock and latch mechanisms. From there, you can inspect the components for any visible damage or wear, and test the operation of the lock and latch mechanisms manually. If you are not comfortable performing these tasks yourself, it may be best to consult a professional mechanic or locksmiths Rotherhithe who has experience with these types of issues.