This issue does seem to exclusively affect the "40" 190TDI Qualltro from comments i've seen here and elsewhere.
I used to have a 2018 Polo GTI+ that uses the same EA888 3B engine as the "40" TFSI unit (but set at 200ps / 320Nm), mated to the 6 speed wet clutch DSG box and there is zero lag/delay from a standstill to the car pulling away in 1st.
I changed to a 2019 A4 Avant 40 TDI S-Line and the reaction from the drivetrain seems to be an age, meaning I have to prod the accelerator a second before I actually want to move.
The Polo was the first DSG box i'd had and I was amazed at how much feathering in of the gears the clutches do between changes, either in the name of smoothness or clutch longevity. The a4 40TDI is built for comfort rater than outright performance, it is likely feathering in the clutch as much for comfort as it is for clutch longevity, and when you put a DSG box in manual mode, there's a little less feathering in in the name of comfort
Although the S-Line is one of those sheep-in-wolf's-clothing trims, it doesn't even try to pretend to be a performance model with driving modes etc - The Polo has those fake "Sport" modes whereby all they've done is dull the throtte response in "normal" and more so in "Eco" mode (with coasting feature enabled also), no fake noise Soundaktor on the A4 40 either.
The "40" 190TDI engine suffers a lot more turbo lag than the "40" 190TFSI engine, due mainly to how much of the total power is attributed to the turbo. VAG used to offer a 2.0SDI (naturally aspirated variant of the TDI lump) that only had 75ps. From that i'd say the 190TDI has probably only got 30-40ps accessible at 1700rpm, before the turbo is fully spun up. VAG has been making 2.0FSI units with 150ps (as featured in the EOS), so from the get-go the TFSI units suffer very little turbo lag vs the TDI. I used to have a manual MK7 Golf GTD which used the same EA288 engine as the "40" TDI unit (in 184ps variant) and even with the manual box rather than a DSG feathering in the clutch, it had a bit of lag.
The Quattro system might also have a part to play. When you have a 2WD car and give it some throttle from a standstill, the wheelspin/tramping acts like a clutch, it's not sending all the power generated straight through the wheels into forward movement until the tramping dies down.
With the Quattro you have instant grip and forward movement once the DSG/S-Tronic clutches are fully engaged, all that power going through the drivetrain to the wheels - that would be a lot of stress to instantly put through the drivetrain if the DSG clutches didn't feather it in. Also, the "40" TDI unt has 25% more torque than the "40" TFSI unit, so there'd be a lot more stress going through the drivetrain on a quattro without feathering in. - perhaps then the TDI has more feathering in (and clutch engagement lag) than the TFSI unit
So the very noticeable lack of response from a standstill as seen on the 40TDI Quattro could well be a culmination of all of those factors:-
1. Higher TDI torque = longer engagement time of the clutches than TFSI to protect the clutches.
2. Lower pre-turbo power available on the TDI and more turbo lag
3. Quattro system requires even longer clutch engagement time to avoid straining the drivetrain.
If all of those contribute, you are going to have a longer 1st gear engagement time on a 40TDI Quattro than other variants.
Longest > Shortest engagement time would likely be:-
40TDI Quattro > 45TFSI Quattro > 40TFSI > 35TDI > 35TFSI
