Speed limits in Richmond Park, Bushy Park, The Green Park, Greenwich Park, Hyde Park, St James’s Park, and The Regent’s Park are governed by The Royal Parks and Other Open Spaces Regulations 1997 (as amended up to 2020), not by byelaws.
https://www.legislation.gov.uk/uksi/1997/1639
The speed limits are laid out in Schedule 2 of The Royal Parks and Other Open Spaces Regulations 1997 and refer to vehicles, not motor vehicles or mechanically propelled vehicles e.g.:-
"Acts in a Park for which written permission is required
4. Unless the Secretary of State’s written permission has first been obtained, no person using a Park shall—
...
(28) drive or ride any vehicle on a Park road in excess of the speed specified in relation to that road in Part II of Schedule 2 to these Regulations;"
...
SCHEDULE 2
THE DRIVING AND RIDING OF VEHICLES ON A PARK ROAD
...
PART 2
Speeds at which a vehicle may be driven or ridden on a Park road
1. On a Park road in Bushy Park, The Green Park, Greenwich Park, Hyde Park (other than the Serpentine Road), St James’s Park, The Regent’s Park or Richmond Park, at a speed not exceeding 20 mph.
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. On the Serpentine Road in Hyde Park, and on the Park road from Kingston Gate leading to the Home Park Golf Club in Hampton Court Park, at a speed not exceeding 15 mph.
4. On a Park road (other than one mentioned in paragraphs 1 ... or 3), at a speed not exceeding 10 mph."
NB An amendment to the regs in March 2010 had included the definition: '“vehicle” means a mechanically propelled vehicle intended or adapted for use on a road.'
https://www.legislation.gov.uk/uksi/201 ... ion/1/made
However the definition was revoked in the November 2010 amendment:
https://www.legislation.gov.uk/uksi/201 ... ion/2/made
Edit: NB for the avoidance of doubt, I'm not saying that the Royal Parks speed limits apply to cyclists. Indeed the current position is that they do not. The police do however enforce a regulation stipulating that you must not ride in a manner that would endanger the safety of other park visitors or yourself, and speeding can form part of that.