This system is tough to diagnose without a factory VW scan tool, but I can give some pointers on where to look:
The first place I would check is fuse #17. It is behind the trim panel on the driver's side under the dash.
Next there are multiple electrical connectors that run voltage to the system. I would probably just unseat each connector under the driver's side junction block, check for corrosion or broken wires and then plug them back in.
Next I would follow this diagnostic flow chart from VW to test the ignition J329 Relay, but again, you'll need a VW scan tool:
1. Access the Voltage Supply Terminal 15 (B+) Relay J329. This relay provides the switched (key ON and CRANK positions) power.
A. Pin 30 at the relay should have B+ at all times. This power is coming from fuse SD5.
B. Pin 86 at the relay should have B+ at all times. This power is coming from fuse SB39.
C. Pin 87 at the relay is the output of the relay. This is where the 15 circuit receives power. Put a jumper wire between pins 30 and 87 and see if the dash will illuminate and the bulb check will happen.
D. Pin 85 at the relay is the ground activation wire. The Access/Start Control Module J518 provides the ground to operate the relay. Manually apply a ground for testing purposes to this pin with the relay installed. See if the vehicle operates correctly.
2. If all of the inputs were present and bypassing the relay brought back operation, the relay has failed and would need to be replaced.
3. If the ground activation signal was missing:
A. Verify the wire integrity between the J329 relay at pin 85 to the J518 module at pin T81/58 Brown/Red (BR/RO). Make sure there is no open circuit, short to power, short to another wire or high resistance on the wire.
B. Verify the power and ground into the J518 module.
i. Pin T81/4 Red (RO) and Pin T81/5 Red (RO) should be B+ at all times from fuse SB41.
ii. Pin T81/1 Brown (BR) and pin T81/2 Brown (BR) should be a good ground at all times.
C. A factory compatible scan tool will be required to verify the ignition switch inputs to the J518 module.
i. Select Address Word 05-Acc/Start Authorization.
ii. Select Function 08-Read Measuring Value Blocks (MVB).
iii. Enter Display Group 001. The 1st and 2nd Display Fields are for the 15 circuit. They should say "ON" with the key in the CRANK and ON positions. The 3rd Display Field is for the 75X circuit. This should say "ON" with the key in the ON position.The 4th Display Field is for the 50 circuit. This should say "ON" with the key in the CRANK position.
D. If the proper power and ground are present and the ignition switch inputs are present, then the J518 module has failed internally and is the cause for concern. If this is the case, this vehicle is a Gen. IV immobilizer system and would need to go to an authorized VW repair facility to have the module replace, coded and adapted to the vehicle with the factory scan tool and access to the FAZIT data base in Wolfsburg, Germany.
E. If the ignition switch inputs are not present, further testing of the ignition switch system and J518 module would be required using appropriate wiring diagrams.