Checked or replaced the HPFP cam follower recently?
Fuel filter change every two years?
You should hear the a faint pump noise for a couple of seconds then stop when the drivers door is used.
It will also prime on the key just prior to holding for ignition.
Have you tried double priming to see if that improves the cold start. I noticed it helped on mine and discovered I had a leaking bleed back valve on the pump due to its age and miles. I got a genuine second hand one as a new genuine pump was £270 + VAT
The latest part number is 1K0919051DB and got one off a 1.4TSI Scirroco 2018 with low miles and 14 years younger for peanuts (£31) to give it a go. I also discovered I had a pinched pump to tank seal from factory like a few other people
Yep new cam follower, old one was fine, cam is fine. New fuel filter too
I have a Citroen Berlingo 1.6 16v petrol that is very noticeable when it primes, both ignition and opening the door, but I don't have that in the GTI, but I don't have one to compare to so wasn't sure if the GTI was just quiet. I can hear a very faint something, but nothing like in the Citroen.
Yes double priming helps - it doesn't always completely cure it, but it's much better after priming it beforehand than not.
I'll look towards the LPFP then - how did you find out yours had a faulty return valve - did you test with a multimeter?