New 20 MPH speed limit to also apply to bikes in Southwark

It's not a byelaw. It's "The Royal Parks and Other Open Spaces Regulations 1997" act:-

http://www.legislation.gov.uk/uksi/1997/1639/contents/made

(Note that's the original act, there are a whole bunch of amendments to it...)

True. The part about no speed limits for bikes still applies either way.