I've commented on your mk1 before a while back as it's similar to mine in looks, same kit/colour. Never going to sell mine... NEVER!

It depends on the system, I didn't just bin everything Clifford related, I binned all the MK1's looms too and started again. It was just beyond help.
If you only had an immobiliser in there, there wont be much to remove. If it's a full alarm system then there could potentially be loads going on behind the dash. With dash and carpets removed I found:
3 brains
Microwave sensors
Glass Tamper sensors
Tilt sensors
Proximity zone adjustment box
Central Locking system
a sh*t load of relays and obviously all the additional wiring that was required to make it all work.
Did you just remove the immobiliser or did you remove it and ensure the ignition wiring was back to how it was originally?.. Immobilisers often have a cunningly placed relay or 2 connected to them to cut the power to either the coil, fuel pump or both.