In the unlikely event that anyone was wondering about the cause of my one-ring-at-a-time cooker (or even remembers it), the reason became clear today.
Before the new cooker was due to arrive, I disconnected (bayonet) the old one and manhandled it out to the front of the house ready for the council to collect tomorrow. Before abandoning it to its fate, I stripped it of any handy-looking parts for my bits box (and I can see an epic barbecue complete with rotating spit attachment being molished later this year after a suitable tuit delivery). With no particular purpose in mind (got a new one for the new cooker), I took off the flexible hose too. Eureka. On a par with the rest of the work in the house[1], the connection from hose to cooker was a textbook bodge, sealed with what appeared to be bath mastic. A goodly gob of said goo was obstructing the hose, with only a pinhole through it for gas to actually reach the cooker.
I can now report all rings burning nicely (and simultaneously) on the new cooker.
Pete
[1] Today's first discovery: on pulling the (all-gas) cooker away from the wall, a set of live electric-cooker connections (bare cores) smiling up at me. Good job I never shoved the cooker back towards the wall too hard, isn't it?