My thanks to Robert, who as always has pointed to the evidence satisfying my initial query. I had muddled the speed constraint with the criterion of being equipment designed for use in construction, but he has put me right. The chance observation that my digger (which is also insured for the minimum requirements for use on a public road) is also exempt from vehicle approval on the basis that it is a tracked vehicle is a delightful bonus. Having browsed Martin's links (and a few other worm holes) I now think I see my way through the bureaucracy. Thank you both.
SOD's observations about driving on a public road are useful, in that to drive on a public road the vehicle must be roadworthy, it must be insured (for the use to which it is being put), and the appropriate tax must have been paid. So my digger is OK to drive on the road as I have third party insurance meeting the needs of the relevant Road Traffic Act, but I must not dig holes while so doing as I have no public liability insurance (I was quoted £2000 a year last time I wanted to dig the footings for a garden wall).
The tax class which applies in these instances is such that the tax is zero (though neither is exempt from the requirement of being taxed). Whether the insurance is invalidated by virtue of the vehicle not displaying a registration number is debatable; of course I was endeavouring to close this element of the debate down. This begs the question of what is a registration number, if the vehicle is uniquely identifiable then I would contend that the law is satisfied. My insurer, by virtue of being happy to issue a Certificate of Insurance showing the equipment serial numbers seems to be OK with the issues arising from the vehicle not displaying a "normal" number plate. If they're happy I'm happy. However I still have rows with my neighbour who is convinced he knows better.
There is also the potential for confusion over what rules apply in the special case of construction machinery being used within the confines of work controlled under the CDM regulations; this does not apply in my case so the point is moot.
As for driving an aircraft tug on the public highway, good luck with getting that insured! My suggestion is that such a vehicle is not roadworthy for at least two reasons I can see immediately, being excess axle loading and out of gauge wheelbase.
The knotty question of young persons driving agricultural machinery is another can o' worms.
Rgds Simon