Depends whether you think Downing Street is an ordinary, public, road ! It isn't. Otherwise anybody could go,without let or hindrance, down it. The gatekeepers, the police, are there to stop that. And it's up to them whether they allow someone in or out, and on what conditions; in this instance, they decided to direct the cyclist to pass through the side gate, a decision which makes sense, for security. The police could also say that they have a power to direct traffic on an ordinary public road too, which presumably includes a power to direct different types of traffic differently!
Seat belts? If the driver without a seat belt on drives out and down Whitehall he's certainly committing an offence. That's plainly a public road to which the law on seat belts applies.