I would guess the buttons are 'proximity' type, with no actual contacts.

The switch pack seems to communicate via the HS CAN via the Switch Pack Centre Console

First find the plug and inspect it and reconnect it, and then check for good Power supply and Earths for corrosion.(most likely both Power and Signal are there)

Repair Manual
https://www.dropbox.com/s/n0e6q5jj1l1fe2z/...6-2010.pdf

Earth Point Positions
 FL2 XS SD4 Auto 2010 2012-2017 (21k - 91k miles) (MY2011)
FL2 Metropolis SD4 Auto 2014 2017- (16k - 82k+ miles) (MY2015)
Metro in its 11th Year of (Extended) LR Warranty / Full LR Service History
(Expensive, but Trouble/Worry free - hopefully?)

I have added Repair Manual Link to post above. FL2 XS SD4 Auto 2010 2012-2017 (21k - 91k miles) (MY2011)
FL2 Metropolis SD4 Auto 2014 2017- (16k - 82k+ miles) (MY2015)
Metro in its 11th Year of (Extended) LR Warranty / Full LR Service History
(Expensive, but Trouble/Worry free - hopefully?)

Thanks for that, I would think that, whilst some of the repair manual is relevant, the trim on the 2013 model (mine) is probably different.
Still can’t get at the switch panel underside.

Have you tried lifting the alloy surround to the gearshift gaiter. On the manual FL2 it’s clipped at the 4 corners. The auto is probably the same.

Sorted!
Found a video showing removal of bright trim around shift lever, actually as stated above, by BossBob:
?si=hOpuTsSEutyjw2va
Then, as with all Freelander jobs, you need to be a bit of a gynaecologist, there is a small green plug underneath the button module, this was unplugged.
It was necessary to remove the yellow(ish) plastic bit at the base of the shifter to give a little more wriggle room. Plug back in, all appears OK.

Glad I could help. Well done for sorting the problem with photos!

Thanks, a very valuable and informative video.
Although it does look that if you haven't got 3 hands, the help of a second person is needed. FL2 XS SD4 Auto 2010 2012-2017 (21k - 91k miles) (MY2011)
FL2 Metropolis SD4 Auto 2014 2017- (16k - 82k+ miles) (MY2015)
Metro in its 11th Year of (Extended) LR Warranty / Full LR Service History
(Expensive, but Trouble/Worry free - hopefully?)