Now that gravity has overcome the forces of my gyroscope, I can tell you that the blond-haired girl riding for Ken Fox's 'Wall of Death' is called Kerri Cameron. She also rides a go-cart within the wall on occasions.
Peter Catchpole was a rider on the wall of death, he lived in Rake Lane. If you Google his name you will get more information.
A friend of mine says Peter Catchpole was the owner of it and also had a small touring circus. One of the riders was called ???Roach and he let my friend have some goes on the wall of death when he was dating Roach's daughter.
I am now able to answer my own question, following a trip to Stafford last weekend when I watched the 'Wall of Death' in action and spoke to the owner, Ken Fox, who is a man who should know.
Ken operates one of the two remaining companies that still tour with a 'Wall' in the UK, and is the owner of the one seen in use on the BBC One station ID. For a while, Ken was a buisness partner of Peter Catchpoole.
One of the things they did during their partnership was to construct a completely new Wall in one of the sheds at Lairds. This Wall is 28 foot in diameter. It was made that size to allow easy transportation while touring. It is the one that was at Stafford last weekend, and is the one seen on the BBC One station ID. This Wall has also been to New Brighton several times for the Wirral Show. (Ken was rather surprised to learn that the show was no more.)
The Wall used in the Tower Grounds until 1959 was 32 feet in diameter and was owned by Peter Catchpoole (and possibly others before him.) It did tour occasionally. I was able to confirm that part of the staging, where people stand, did collapse in 1959. The Wall was taken out of service, repaired and extensively modified, but didn't return to New Brighton. It was taken to Africa, where it toured, ending up in North Africa.
Peter Catchpoole, a former rider himself, went out there with it, where he set up some more enterprises, particularly of the fair ground variety. Peter ended up in Nigeria before returning home. He ran his buisness from his home in Rowson St, New Brighton, and died in 2002.
There is a variety of information on the internet about the Wall when it was at the Tower Grounds, and about the other buisnesses of Peter Catchpoole. A number of riders names come up, some of which were real names, and some of which were stage names. It seems likely that another Wall did appear occasionally at the Tower Grounds during the 60's, but it was not the same one that partially collapsed in 1959.
Thanks for the link, as I had only seen some stills.
Here are some other things that I have found out (from the internet)-
There was a female rider called Ursula Brown (nee Cole) born in 1933 and died in 2011, as in one of the earlier links. Another female rode as Ingrid Johansson, but her real name has been given as Ada Roberts. Freddy Heywood and his wife owned the dodgem cars and may have also owned and ridden the Wall at some time. Another owner has been given as William 'Ron' Miller. One of his riders was Nobby Dare (Clarke) and he rode with a monkey. Later, Peter Catchpoole and Ronny Swift were riders, along with Johnny Bates.
One or two of those names appear on Ken Fox's Wall as riders who have passed on.