I have road bike and have cycled across Wimbledon common loads of times. Most of the paths are suitable for thin tired bikes as they are not too stoney and bumpy. The only problem is mud so fit mudguards to your bike if possible. You may struggle to get up some of the steeper hills if traversing the common from east to west but being a road bike they are light enough to pick up and carry or push up hills.
I have road bike and have cycled across Wimbledon common loads of times. Most of the paths are suitable for thin tired bikes as they are not too stoney and bumpy. The only problem is mud so fit mudguards to your bike if possible. You may struggle to get up some of the steeper hills if traversing the common from east to west but being a road bike they are light enough to pick up and carry or push up hills.