Clogzz wrote: I start my car twenty minutes before departure
Wa-wa ? What car ?
Is that a vintage steam-driven car ?

Good gracious, I never realised it got so cold in Sydney that you needed to do even more than we need to do in Blighty even in the dark depths of winter Adam

It's not as if you live in Canberra - up there I could understand leaving it running for 20 minutes or so before setting off in the winter...
Back to the original question, yes, as Adam says and remember that the AL4 is auto-adaptive and will learn your driving style to the degree that if it understands you like to press on a bit then it will hold onto gears longer. It will do this when cold on purpose to assist warm-up of the cat and transmission oil and will do it moreso if the sport mode is engaged. You may have sport engaged and be unaware of it...
If the tacho is dead, this may be having an effect on the autobox as it reads engine speed to help determine when to change. A Lexia diagnostic session will be helpful in determining the cause of your dead tacho; I'd get that fixed before anything else...
Whats the mileage on your car now? has this behaviour only just started or has it always been there?
Also, if the car has done a reasonable mileage and you have no record of the transmission fluid being changed, it's not a bad idea to change it. Not a straightforward job as you need to follow a procedure and you must use a Lexia diagnostic machine to ensure the 'box is at the right temperature for level checking and to resent the oil wear counter.
Also, with a Lexia, the auto-adaptive parameters can be reset, allowing the ECU to re-learn the characteristics of the 'box. It might be worth doing to get everything back to normal but fix the tacho first.