An update. I deflated the tire down to flat and gave it a good pinch and massage all over before pumping it up about 15psi at a time. When I got it to about 60psi, I spun it and it certainly looked better. Then I decided to let it down again, remove the tire and place the rim on the bike tireless. The rim is as true as the day is long. So I put the tire back on following the same procedure as last time. This time, I happened to notice that the wall area on some parts of the tire close to the bead looked a bit battered. When I fully inflated it, I spun it and bugger, the wobble was back.
So I took the tire off again and swapped it with an old but identical tire (Continental Travel Contact) and when I spun this, no wobble. The wall area close to the bead also looked in much better condition. So tomorrow, I will go for a quick spin and try to find a smooth downhill in the 20 minutes I'll have before work. I am feeling confident I have nailed it and my tire was cream crackered. I'll update you again later in the week.