Thanks for the advice Max and Norm. Yes it is the ferrules that are the problem, plus finding a cable stop cap that is big enough and strong enough to stop the cable popping out when you push down on the lever. I've since learnt that as these throttle controls are made for vintage motorcycles, namely BSA, Triumph, and Norton, if I buy a cable for one of those bikes it will fit, perhaps with some alteration on the carby side. I have been buying universal lawnmower cables, which is why I have been having so many problems with things not fitting right. Max, that first ad you showed won't fit. I know because I bought one one time and it would only fit if I cut the lead piece down, and then I couldn't get it to stay in the control because the stop ferrule was not right either. The other two you showed might work though.

Also, I don't know of any shops that still fit throttle cables for customers. I have asked in a few lawnmower and motorbike shops locally and they just shrug and tell me to buy the modern universal plastic ones, already fitted in the packet. I am sure they would have even less of an idea about how to fit these old Amal cables than I have. So if you can suggest such a shop by name and address, I would be very grateful. I've searched online but no luck.

Incidentally, I knew a bloke years ago who was a top mechanic who could engineer almost anything in his backyard workshop, but he always said "Don't ever bring me a throttle cable!" He could do most things but he considered throttle cables the work of the devil. He hated them with a passion. He would not even try to fit or repair one, but always left that job to a friend of his who apparently did have the knack of fitting them.