Spare Wheel Removal

You shouldn't really need a different socket, the wheel brace should fit over it,though it might need a little persuasion from a hammer but only very lightly taping. You ought to be then able to just twist and pull and get it off or just unwind it completely.

Is there enough space to get something down the sides of it like a long thin screwdriver?
I was thinking if you had something like a long thin screwdriver and bent the tip of the screwdriver over so that it could catch on the furthest edge of the offending cover, and pull it towards you. Obviously you will be pulling on one side, so may need to work around it until it starts to shift

I cant think if anything else unless its magnetic and you can get a strong earth magnet to it - Don't lose the magnet down the hole too!

If you've established that this is just a tin cover on a proper nut, it's likely rusted on and I would be inclined to go at it with a screw driver and hammer via the hole drilled to rip it open and loose. Then you will have created an edge to be able to get a grip on and pull. It's no good to you as it is and the gently gently approach is failing. Just be careful not to damage the plastic cover.

Kirkynut

Thanks for all the suggestions folks. I'm new here and have been really impressed by the level of response from so many other Jimny owners wanting to help.

Unfortunately 'Jim-Bob' and I are having an enforced separation for a couple of weeks so I won't be able to act on any further suggestions, or report back on progress for a while.

No worries, hope you have a good time whatever you are doing.
But don't forget to report back here with any more ideas that you've had or things that you've tried, and let us know the outcome... if any!

I'm intrigued now and want to know how it pans out

From what i can see, that would appear, as already said, a security bolt of some sort..May i suggest a steel drill shaft pushed into the hole, and then a ring spanner, around it(slim one)..push the drill shaft into the hole firmly, while trying to undo the nut..
I have seen similar security nuts before..