My headlamps don't always pop up which is probably a very, very common issue..
I'm trying to figure out which part to replace.

When they actually do go up, they do it at the same time and absolutely perfectly.
But sometimes they just stop working and it can take anything from 5 seconds to 5 days of driving before they suddenly decide to go up again.

If it was a motor issue, wouldn't they always work badly and you would also expect situations where only one of the two lamps goes up?

I have replaced the switch, no effect there.

 

Could be a bad connection or loose or broken wire. The issue I found on mine was a bad ground wire on the driver side tucked up under in front of where the battery is. Took a long time to find as it was out of sight. Go through the wiring and check each connector is clean etc. etc.