After researching this, including enlisting a few friends over the last couple of months, it seems very hard to find towing law in one place. Many different sites contain advice and most of it is conflicting.
Anyway - on a site with technical advice from Warwickshire police and links to the relevant legal documents I've found something that helps shed some light on this murky area.
Using an A-frame within the 'trailer' rules: In UK law, the car being towed can be classed as a trailer - nothing more complicated than that. However as the trailer (towed car) will not be using it's brakes, it is classed as an 'unbraked trailer'. The legal limit for weight of an unbraked trailer is 750kg, therefore a complete Minor 1000 is too heavy to tow as a trailer. If you reduce the weight (e.g. removal of engine and gearbox = 641kg) you are below the 750kg limit and you are legal to tow the Minor as a trailer using an A-frame... but the weight must not be above what is plated for your tow vehicle (or 50% of the tow vehicle weight, whichever is lower)
Most tow vehicles (e.g. family saloon) are not rated for over 500kg unbraked trailer so this is your limit. You can go for a bigger tow vehicle (with bigger allowance for unbraked trailer) or reduce the minor's weight even further (reaching 500kg is not likely to be an option in most cases)
Using an A-frame for emergency recovery: In this case it is ONLY for emergency recovery - i.e. to move a vehicle to a safe haven. This would be the nearest town / garage / service station only! under the allowance for emergency recovery the rules are far less restricted, however if you go past the nearest 'safe haven' then you are outside the rules.
In either case the towed vehicle needs a trailer board at the rear to show lights and registration of the towing vehicle.
So with a heavy duty tow vehicle you can legally A-frame a Minor 1000 if the engine and box are removed

Towing dolly
These are like a mini-trailer which lifts up one end of a car and the other end of the car remains on the ground.
The same regulations apply as when using an A-frame. Some dollys have a braking mechanism which of course is much safer (if designed and working correctly). However as this will only brake the dolly wheels and not all the wheels of the trailer (ie not the 2 car wheels running on the road) it can not be classed as a braked trailer and therefore legally no different from using an A-frame.