I have a three button fixed key; one button locks, another unlocks and the final one unlocks the tailgate. Last night the key would neither remotely lock nor unlock the car (strangely the tailgate button still worked) so I had to put the key in the door and turn it. I can put the key in the ignition and start the car without any problem.

At first I thought the battery may have gone, so I replaced it with a new CR2016. Still the same. Then I tested the spare key - exactly the same problem. Now neither key will unlock/lock the car.

I'm scratching me head to think what could have gone wrong?

Sounds like the key has lost sync wth the car, there is a way to re sync them but I'm not sure what it is, though I'm sure someone on here does!
This may help.
http://www.fullfatrr.com/forum/topic3713.html?highlight=key+sync There is nothing that can't be fixed with a hammer😜😜
FFRR 4.4 SDV8 Autobiography Santorini Black.
KIA E-Niro 4+
2010 LR D4 Commercial 2.7 TDV6

Your Bcu has gone into transportation mode, don't ask me how or why they do, but they do sometimes.
When we got our motor it was the same, only the tailgate button on the remote would work,
You might find other things don't work as well. Interior lights, central locking, tailgate release from the outside to name a few.
A good diagnostics kit should sort it.
Three that I know will sort it are Autologic, RSW All comms and Faultmate.
Someone might also confirm on here that the Gap IID unit will as well.

Steve. 1968 Series IIA Pastel Green, nearly there project

Hi Rextter,
Vaz is right - they do go into transport mode every now and again. Has your dome light stopped working when you open the door? Then you are in transport mode.
As Vaz says, a good diagnostic tool. And yes, the IIDTool will turn transport mode on and off.

Just slightly OT: Actually at this point there is very little the IIDTool is not able to do. We have not yet released re-flashing capability - this will come shortly on the IIDTool Pro. We are then definitively up to Autologic/Faultmate level of capability. People still think of us in Hawkeye/Lynx terms because of our price... 
RRC 2Dr, RRC 4Dr,
P38, and 2 L322s
(wife thinks I'm nuts - prob right, too)

A very nice mobile diagnostic man visited me today and plugged his computer into my car. The BCU (Body Control Unit) displayed the 'Transportation Mode' error message which he was able to clear in seconds and my lovely Range Rover was back to normal.

Thank you to the good people of this forum for pointing me in the right direction!