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I'll bite. I think that quite possibly speed limits do apply to bicycles in Royal Parks. At best it is ambiguous.
We all agree that the original Regulations used "vehicle" which, in normal use in legislation, includes bicycles. We all agree that the amendment regulations used the definition "mechanically propelled vehicle" which doesn't. I think it is ambiguous whether the new definition applies only to the places where the amendment regulations insert changed text into the original regulations (which is primarily to do with car park charge iirc) or to all places where "vehicle" is used. If the former, speed limits (probably) still apply to bicycles, if the latter, they don't. The consolidated regulations, which are not I think definitive but may be indicative of how lawyers interpret it, opt for the former iirc (I'm on an iPhone so doing this from memory).
Then there's the policy statement that you helpfully pointed me to when I asked previously and which Tim linked to upthread here. Firstly, a policy statement can't rewrite the law (a parallel with the Highway Code). Secondly, I'm not persuaded that the statement of policy in that document is any more authoritative than the other extant statement of policy in the shape of the notices in Richmond Park which say that the limits do apply to bicycles. Thirdly, I'd be happier if they hadn't muddied the waters by talking about design guides and 8-12 mph in the same paragraph, which surely apply to off-road not road cycling.
So my conclusion is that I honestly don't think there is any certainty as to whether speed limits do or do not apply to bicycles in Royal Parks.