2003 Volkswagen Golf intermittently loses power on start up - usually when cold and or rainy

Is that just as you start it in the morning? Once you have got it going its fine all day?
Maybe worth checking the fuel pressure to see if it is draining back overnight, or could be a leaking injector dripping fuel into the chamber making it choke with too much fuel untill its cleared it by revving it.
I'd also say maybe the lambda sensor but its not used until it is hot(closed loop) iirc. But worth disconnecting it before you start it to see if its better.

You say it doesn't throw a fault. Do you mean no light comes on or nothing is logged? Have you access to a vag. Com reader to check, try also reading the live data, see what any of the readings do when it messes about.
As to why it does it when cold or wet, I'm a bit stumped otherwise unless it has a sort of choke(cold start) read from the temp sensors?

Thanks. Bit over my head but will suggest to a mechanic. I left this a week with a garage and they ran tests on it and no fault codes show up. Someone has also suggested a faulty temperature gauge at start up. I'll need a savvy mechanic and put your suggestions to him. It's very frustrating when you know nothing about cars. Thanks again appreciate your suggestions.

You welcome mate hope it helps Sometimes its better to know nothing and pay someone professional to fix it, trust me lol. Some faults can drive a mechanic Insane😈