The Tufty Club's road safety campaign started up in 1961. To help promote it they had Tufty teams visit schools, including mine, Balerno Primary, and dished out badges & booklets.
The Tufty Club has another link with Dundee, although not related, this being the name that was given to an under 14's disco they had in the early 70's at the JM Ballroom. I never heard of any kids being run over on their way to the disco, so at least all that 60's road sense paid off!
I remember even into the 80's when there was an under 14's disco at the Barracuda on Sunday afternoons, that too was referred to as the Tufty Club, but I think by then it was a patronising term spouted by older clubbers, rather than by those who went there.
And as a wee reminder of what triggered it all off, here's the original tv ad below.