Generally paths that have been paved with sideways paving across the path are designated for walking and with parallel paving are for cycles. Quite often you will see half and half. Its not always clear or signposted what that means (and different councils tend to have differing implementations) - however common sense prevails. If you ride at walking pace (and its hard not to) through a cut through like the Shipman Road/South Road one, then you're not doing any harm and there is enough room on that path for everyone.

I wouldn't usually ride my bike through there, but have no problem if others do. There are a lot more ridiculous cycle/foot paths around London!