It's great that every random person with a digital camera thinks they have the right to call themselves a "filmmaker". You're not.
Regarding the right to video someone... standard laws apply not to the actual act of videoing, but to the financial exploitation of said video. If you video someone without their permission and then attempt to financially exploit said video, the person appearing in the video can sue you and force the video to be destroyed. The same goes for recording any and all intellectual property.
This, for example would apply to the crests that appear on the police helmets. Assuming it's the same in the UK as in New Zealand (which it's most likely it is) the copyright to the crest on the helmets belongs to the UK government, and as such videoing and broadcasting that crest without permission from the copyright holder constitutes copyright infringement.
The laws regarding documentary film are a little vaguer as it can fall into the news gathering area where copyright is exempt, however any actual filmmaker is aware that they need to get personal releases signed for any individual who appears in their film.
This can be avoided by blurring or concealing the face of the person being recorded, which is a common tactic used when the subject matter is likely to refuse to sign a release.
Incidentally, if UK law is anything like NZ law, crossing a police barrier constitutes a very serious offense, and the videographer in question is lucky they are not facing charges.