With certain very limited exceptions (relating to matters as varied as the Official Secrets Act, indecent images of children and CCTV on commercial premises) anyone is free to photograph (using either still or moving images) whomever and whatever they like as long as they (or their cameras) are located:
(a) on their own property ; or
(b) on property where they've been granted permission for photography ; or
(c) in a public place.
Even if they take photographs (or record videos) elsewhere, it's only a civil matter, not a criminal one.
So, for example, it's perfectly lawful to film passers-by from your house. Similarly, you can stand in the street and film people in their front garden or (through their windows) inside their houses. Even if you walked into their front gardens, and continued filming them from there, you would not have committed any criminal offence.
So it follows that it's perfectly lawful to film whatever is going on around you from within your own car, which (as it's on the public highway) counts as being in a public place. It could technically be argued that you would need Asda's permission to continue filming once you drove your car into their car park but
(a) plenty of people take photographs inside Asda anyway (with no problems, other possibly than being admonished by the security staff) ; and
(b) as it's only a civil matter, Asda would find it difficult to sue you because they'd not suffered any loss through your actions.
When street disturbances (or other public visible offences) occur it's now fairly common for people to reach for their cameras (or mobile phones) in order to record the incidents. No court would rule such evidence as inadmissible, so I can't see how evidence from dash cams would be treated in any other way.