I took the seat to a local upholsterers as I thought it would be an easy fix for him as he had the tools, materials and experience to carry out the job. However he didn't seem very interested, although he did say he would have a go, but wasn't confident that he could provide a solution that would last. I was surprised at this and even my wife commented after that he was bothered.
My next plan was to do what Hamwich did and lace the bottom of the seat with something like thin elasticated rope, such as the type used by climbers, only thinner.
I like the look of the link (MGB seat base) that Treesurveyor has posted and think that that could be modified to suit.
Treesurveyor were you able to drill the holes in the frame with the seat cover still in place or does it have to be removed to drill them.