Likely a cable problem...  that could be the result of a speedo Locking up .
   Check the cable both ends. they usually shear off down by the transmisson end.

if sheared. check that the speedo still spins. ( use the old inner cable to insert _ and twist  If it twists it should be OK.

Likely broken cable .. for multiple reasons.

"the speedo magnetic pickups went bad or something interior to the actual speedometer. "
If you have an A+ engine, and a conventional mechanically driven speedo, the only magnetic component is inside the speedometer. The cable is driven by gears in the transmission. In the speedo, the magnetic drive does not physically touch but relies on the magnetic field to turn the needle against the spring's resistance. If the needle is at zero, the speedo is probably OK because the needle is free enough for the spring to return it.

If the cable got too close to the exhaust manifold, it could have been melted by the heat, though that would not damage the core other than make it stick.

It is possible that the core tips may have been damaged or the cor is slightly too short. I ahd that problem with mine - on a drive the speedo would work properly until the outer sheath of the cable warmed up and expanded. This would allow the core to disengage at one end or the other. When that happened, the needle would drop to zero ... Warp Speed!

The cable could have broken because its probably never been lubricated. Take the end off the speedometer and drive or push your car and watch whether the cable turns or not. Or you could pull the cable out of the sheath to check its length.
If you need a new speedometer, here's one for sale which might be cheaper than Mini Mania. I know the seller has several of them. 
https://vancouver.craigslist.org/search/sss?query=austin%20mini%20speedometer&sort=rel

Need some advice for a new speedometer, or repair of old one.  Was driving my 89' Austin/Rover Mini the other day, cruising at about 50 miles per hour, and all of a sudden the speedo stopped working.  When i got home, i checked the connection behind the dash binnacle and that seemed good (had not become disconnected) and looked under the car to see that my cable was still connected to the engine. No problems seemed apparent. My only guess is that the speedo magnetic pickups went bad or something interior to the actual speedometer. Your thoughts on replacing or repairing.....will take any advice, or a known location if you know someone looking to sell one. 

As a side, i have looked into changing it with a new Smith's...but my current speedo doesnt have the gas gauge...and would prefer not to rewire the circuit board or fuel sending unit.