What you describe is the normal function of the alarm/ immobiliser.
Immobiliser is still on if this is the case. It's likely that the bonnet and boot switches aren't connected hence only the doors setting off the alarm.
When you press the unlock on the key fob try getting close to the alarm module in the car.
It's hidden behind the vertical fabric area of the dash on the passenger side.
Sounds like your key fob has developed a fault sadly.
Usual faults for key fobs are around cracking of the solder of the press buttons on the circuit board so have a look when you open it.
Also look at the battery connections and I am assuming here you put the battery cell in the right way around and it's the right kind CR2032 from memory.
The only easy way to check is to get a key fob tester (ebay etc)
FS
The both the boot & bonnet still active the alarm if you open them individually. So, I'm assuming the sensors are still working?
The car has a wooden dash & console (see photo) so where exactly do I try the the fob close to the sensor. I've opened the fob again & everything seems intact (see photo). There are some clips that are holding the circuit board in place. I is it wise to open that too or will that total destroy the fob completely? If it is the fob, are replacements available or will I have to have the immobiliser taken out? Also is there a way of by-passing it so I can get the car started again? Sorry for all the questions, but I'm using it as an everyday car at the moment & with it not running is really disrupting my daily commute.
KeyFob Open - http://flic.kr/p/hAcuQz
Dash layout - http://flic.kr/p/hAbZ16
Okay so your bonnet and boot switches are still in place (they all work by earthing the signal same as the door switches).
Open the glovebox and try inside it at the back right hand corner.
It's the other side if the key fob which has the buttons on, you should be able to lift the circuit board out carefully and check the little button on the board.
The key fobs are a fixed code and the alarm box 5AS learns their code rather than the other way around.
If you need a replacement fob it's a specialist job as the 5AS has to learn the key fob code whilst it's in programming mode.
Removing the alarm box won't help as the 5AS is also coded to your ECU.
There is no way to bypass it without specialist equipment sorry.
It's a pity you are so far away as I have the gear to programme the 5AS to another key fob and the ECU (in case the 5AS is faulty).
There are some specialist companies who offer a postal service or maybe you can find an ex rover garage who still has the equipment (it was also used in land rovers).
But first I would be trying to establish it is definitely the key fob giving no signal
FS