@fimm
OK, my post was a bit confused. If the dashcam is not visible from the driving position then it's OK (per what I said about the barmpot watching TV - same rules). But it still seems pretty pointless. (Usually they revert to "viewfinder" mode when the USB power is disconnected and they're running on the small internal battery. This means that you can use them to take still photos if required after an incident.)
AFAIK pretty much all of the reputable ones sold as dashcams (such as the NextBase range, for example) do turn off the live video display when the vehicle is under way. Some of the cheap Chinese/Russian ones may not do. And some people use a mobile, or a GoPro or some other camera not specifically designed for automotive use as a dashcam, which also won't necessarily turn off the display automatically.
I can't locate the relevant regulations online just now. I may have a more thorough look later.
EDIT: It's in the Construction & Use Regulations 1986:
109.—(1) No person shall drive, or cause or permit to be driven, a motor vehicle on a road, if the driver is in such a position as to be able to see, whether directly or by reflection, a television receiving apparatus or other cinematographic apparatus used to display anything other than information—
(a) about the state of the vehicle or its equipment;
(b) about the location of the vehicle and the road on which it is located;
(c) to assist the driver to see the road adjacent to the vehicle; or
(d) to assist the driver to reach his destination.
(2) In this regulation “television receiving apparatus” means any cathode ray tube carried on a vehicle and on which there can be displayed an image derived from a television broadcast, a recording or a camera or computer.
Note that "other cinematographic apparatus" means that it doesn't just apply to CRT screens, despite what some people seem to want to argue based on paragraph (2).
There's also a school of thought that argues that a live dashcam picture is displaying information "about the location of the vehicle and the road on which it is located". Same as the view out of the windscreen, then... And AFAICS the only way that a single unit dashcam could usefully do that would be by being located in contravention of the rules about obscuring the windscreen. I'm not aware of it having been tested in court, but the informed advice seems to be to avoid it unless you really like funding lawyers' lifestyles.