The Richmond Park 3-Lap Challenge

Not true in Royal Parks.

On normal roads the speed limit only applies to "motor vehicles".

But the Royal Parks Act is worded slightly differently and so it applies to all vehicles (pedal cycles included). So you can be charged with speeding on a bicycle where the Royal Parks Act applies (this includes Richmond Park).

Not knowing your speed isn't a valid defence either.
It's been discussed upthread a fair bit - The speed restriction probably does not apply to cycles, but this has not been fully tested, as everyone appears to have paid the FPN.