This may be a daft question, but why would the reverse idler gear have any involvement in the selection of only two forward gears? It may well have its own problems, but affecting 1st and 3rd only? How?
How old is this Mini? Pre rod change, this is the exact symptom of worn, or loose on their splines, knuckles in the linkage. Modern rod change, quite possibly nothing more than linkage or even worn roll pin or coupling, or engine not sitting correctly (worn mounts?) so there isn't enough travel in the 1st and 3rd direction.
I suggest not having the engine out until the linkage is fully checked. Hope it is an easy one!
Having read this thread, I highly suspect it's the reverse idler bush on my 1986 A+ City E that's causing the stiffness I've found when changing across the gate. It's really bad and actually gives me a sore elbow if I've driven the car for a long journey.
To put it another way- my mini has a very stiff gear lever moving anywhere side-to-side whilst trying to select gears.
Key phrases bolded 
Being an 86 A+, it's almost certainly a Rod Shift, unless someone has changed out the gearbox at some point.
The stiff movement of the gear stick, from side to side across the Neutral Gate is a symptom of the Reverse Idler Gear Bush walking out as Midas Mk1 has said.
When the bush, which were two piece in this model, walks out, it doesn't allow the reverse Idler to go back completely to the Neutral position, the slots through the Bell Cranks don't line up and this exerts a force on the Shifting Shaft and Interlock Spool, making it stiff to move. As it gets worse, the only gear that will be selectable will be Reverse.
Once the engine is out, it's only a matter of separating the gearbox off it, removing the laygear which gives access to the reverse idler, whip that out, fit another (with a 1 piece bush!), reassemble and then it's time for a well earned beer.
Edited by Moke Spider, 13 July 2016 - 08:33 PM.
