2004 VW GOFL - manual transmision

I have a 04 VW Golf. I bought it new and had the transmission fluid changed two years after, at approx 35000 miles. Since then, only in winter when it is cold, I have trouble switching from first to second gear and back. All other gears work perfectly well. I thought that maybe the mechanics did not put the correct syntetic oil in the transmission and had it again changed a year after that by another mechanic and I made sure that they put correct oil, but it is still having the same issues. Any advice? I would really appreciate it. REMEMBER, it does it only when it is cold, when I first start the car, until it warms up, and runs well after that and in warm times. Thanks! Alma

This sounds like a clutch hydraulic problem with either the clutch master or slave cylinder and has nothing to do with the type of gear oil.

Cold is causing the rubber in the seals to harden (combined with age) and this is in turn causing the clutch to probably disengage as much as it should. When the seals warm up the rubber expands and softens up a bit which will then lead to it shifting normally.

You could try having the clutch hydraulics bled and see if new fluid would help. If not, and especially if this is leading to any gear grinding, then the cylinders should be replaced. Both the clutch master and slave cylinders should be changed since one usually follows the other.
JMHO and hope it helps.

Is the problem a stiffness in the shift lever (it moves smoothly, but slowly like honey vs water) or does it tend to stick.

I believe OK has it, but if it just moves slowly, the fluid is wrong, you may need a synthetic vs standard.

No, it totally sticks and synthetic oil is in there.