What's my budget? As little as possible, but ideally under £ 200. It seems I can get a pocketable camera like the Sony DSC-W830 recommended by
Expatrick for that, but it only does 720p, whereas I'd really prefer 1080p. That seems to put the price up by a factor of two, so I think I need to either save up a bit more or lower my expectations. I've also asked TheLuxuryTravelExpert what he uses, but I suspect it will be some whizz-bang 4K camera with a price tag in four figures.
Thanks everyone for your most helpful advice and suggestions!