An update - I got the speedo cable connected by going down from the top with a bare arm after trying some other methods. It was a squeeze and my hand was pinched and poked but I got it. Thanks, all. 

Last one I did was from the bottom....can't see doing it from the top.
I would caution you to remove your watch first...if using that arm.  I got stuck for a little while...lol

Thanks, Jedduh.

For a really stubborn one .. Remove the CV axle from the Transmission = After draining the oil of course
  I did this to install a  Electronic speedo pulse sensor = and making sure it was all properly engaged.

  Good to also go thru the radiator side with a long screw driver = tap the edge of the screw bit with a screw driver or such to break it loose.   They only need to be finger tight.. Dirty threads help keep it in place.


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 I Can reach down behind from the top but its only fingers . no strength to turn anything.

Thanks, Willie. 

No, get the car elevated and reach directly up from underneath.

Alex, would you separate the drive shaft joints to get at it? 

I've always found it easier from underneath.

Thank you, Dan. I am trying to put a new cable into the transmission / final drive connection and it seems quite impossible.  I will dive in there again with your encouragement. 

Fosterbeagle. 

Some of that might depend on whether the car has a centre binnacle speedometer or a 2 or 3 gauge cluster in front of the driver. Cable types and lengths would vary.
 
If you are just trying to reconnect the cable to the drive on the transmission then things are simpler. If you have long-ish and slender arms, you can reach it from above without dismantling the rad, hoses etc. Tricky and tedious but it can be done. (MY arms are not long and definitely not slender, but I managed.)

What is the correct method or the easiest method, or indeed any method to connect a new speedo cable to the fully installed and otherwise functional RHD classic Mini?
Fosterbeagle