Rule 66 says you SHOULD keep both hands on the handlebars except when signalling or changing gear (plus some more stuff), which means it isn’t illegal to ride one (or no) handed, just not recommended.
The law about not using a mobile phone whilst driving is specific to driving, it doesn’t apply to bicycles.
As I understand the law, it is not a specific offence to operate a mobile phone or radio whilst cycling. There are catch-all offences of “Dangerous Cycling” and “Cycling without due care and attention” (RTA sections 28 and 29) which may come into play if the boys in blue don’t approve of your activities or you are involved in a collision.
The biggest issue I have is actually more practical - wind noise makes both transmitted and received audio difficult to copy if you are travelling at above walking pace. I would personally be happy to operate using a light headset and locking ptt if I could find one that was comfortable and immune to wind noise. I’ve tried some of the devices like the in-ear bone-conducting headset sold by some Hong Kong based eBay sellers, but not been particularly impressed by them.
I have used a bicycle for SOTA (G/SE-002, G/SE-003 and G/SE-004 in one day - 108 miles on the bike and qualifying all three peaks in under 12 hours), but only used the bike as an antenna mount on one summit - and that was with an MFD-style 2m dipole pushed into a pannier to hold it upright, rather than something that would work whilst mobile.
I also tried some APRS experiments on the bike. My sport is Audax and Ultracycling (fast, unsupported long distance riding - for example my favourite annual organised event is a ride from Chepstow, up through mid-Wales and the Elan valley to Aberystwyth, pushing up through Snowdonia and Pen-y-Pass to Llanberis, over to Menai Bridge, then back to Chepstow. I ride to the start and back home afterwards, so cover 600 miles on the bike between leaving home on Friday evening and returning on Monday morning). I thought it’d be good if my family could track me on aprs.fi - if it shows that I’m still moving then I’m still alive. For this I used a 2m antenna securely mounted to the back of the rack. Unfortunately the UK APRS network doesn’t give good enough coverage, especially when operating on low power into relatively inefficient aerials. Cranking up the power isn’t really an option as this means carrying multiple battery packs or heavier high power radios that would interfere with the cycling.