Hello and Welcome..
My name is MIKE - Im happy to help / advise you..
This is . or should be - a Rod change - If you look under the car - from the base of the gear lever - you should see two rods going to the gearbox -
Some had issues at the base of the lever , basically seizing on the joint from the lever to the rod - so you could check that for free movement - - then, the next thing to check on this is that the roll pins that attach the gear linkages have not broken - near the trans..they used to break - meaning that you could go through the selecting process with the gears - but it was just swivelling in the linkage - nothing therefore happening in the g'box.
So you really need it up on ramps - have someone to manually go through the gears ( engine off ) while you watch the linkages - to se if thats whats happened... if so, that can easily be repaired by replacement of the pin.
Other than that - if on checking you find the gear linkages are indeed all actuating through to the g'box- then you have a broken selector internally or faulty reverse gear train ( cogs / shaft )
With these -you would have to remove the engine - then on the bench / ground- split the engine from the g'box... by first removing the large clutch cover - clutch and flywheel - ( you need a special flywheel puller tool on these to remove the flywheel / get it off its taper off the crank
- one such tool is HERE
for your reference.) - - then all the bolts around the g'box top case / bottom of engine - then it will separate for you to then inspect whats happened and go from there.. Its quite easy from there to be honest as access is then easy to the internals of the g'box.
You can get the parts from MINI specialists on the web - they advertise on google etc. I cant tell you the costs Im afraid as we dont know until its stripped just whats wrong - but its not usually expensive in parts - its just labour intensive / time consuming - which is not an issue if you are doing it yourself...
If there is anything more by all means
get back to me and we can carry on... *** >> Meantime >>>>
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in advance for that...
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Kind Rgds, MIKE.