Can members please advise. 

After some years on non use I am recommissioning my 1985 K100RS.

She has a good spark, has new fuel in the tank, new spark plugs, new air filter, new battery, new fuel pump, new fuel filter, s/h Motorworks fuel pressure regulator, all injectors professionally serviced.

Action. Engage "choke" (fast idle), turn key, ignition lights are good, press starter button, fuel pump whirrs away, starter motor turns, the engine starts - and then just as quickly she stops. Repeat as before. Same.

As per above, I've checked the fuel delivery side. All seems fine.

Last night I was watching an old car show on TV. In the program a technician had a similar problem with a fuel injected V8. He concluded that the fault was an airbox flow sensor failing to operate. The computer detected this non return, and because there appeared to be no airflow, shut down the ignition.

In like manner on my K100RS, could the Motronic fuel injection control unit and the ignition control unit be conspiring to produce the symptoms that I above describe?

How do I test the ignition control unit and the fuel injection control unit?

Or am I here barking up the wrong tree?

The only way to test them is to swap them for known good units. Both units rarely give any trouble and an 85 will have a L-Jetronic as opposed to a later 16 valve which has the motronic system.
The injectors are controlled by the ECU but the ICU supplies an earth to the so the pump will run before the engine turns over. I would suspect that this is where the problem is in that when the start button is released the earth is removed and the ICU thinks the engine is not running and stops the injectors firing by stopping the pump.
Keep the start button depressed that way you will isolate where the problem is. Don't be concerned about the starter spinning as if the engine is over 711rpm the starter will not work.

Disconnect the unit under the seat and clean the pins on it too. Electrical/connection issues as opposed to component issues are very high on the list. If you have any K lovers near you [there are number of them on here] a quick swap over of the unit under the seat is a good move for beer money, only takes a couple of minutes and either fixes it or tells you its a good unit by working ok in the other K. All K100 2v ones are the same for that unit under the seat.