LED light bulbs turn off immediately after being switched on
This question is old, but it just hit 1000 views so I thought I'd post the answer. 

The 12V PSUs were all at end of life, and were failing. It may have been that the 3.5W bulbs were too low power to work, but either way I ended up replacing the lot with new sockets and LEDs. If I recall these are not stepped down 12V but run off 240V so don't need transformers. I could be wrong about that, it's over a year since all this happened