Let’s say someone has Saturn at 13º Taurus. If we change this to a sidereal coordinate it would be about 20º Aries. This shows that 13º Taurus tropical is the same place as 20ºAries sidereal. Saturn does not change its position, only our way of describing that position changes.
So, let’s say, on a certain date Saturn comes close to 13º Taurus, tropically speaking. It would also be close to 20ºAries sidereally speaking – because 13ºTaurus topical and 20º ♈︎ sidereal are just two ways of describing the same point in space. This is why it doesn’t matter whether you use sidereal or tropical descriptions, the transits to the various points in space will happen at the same time.
Here is an example to help you visualize it. Lets say I am three feet away from one wall, and ten feet away from another. From one point of view you can describe me as being three feet away from a wall, this is analogous to saying “Saturn is at 13ºTaurus” from a tropical reference. From another point of view you can describe me as being ten feet away from the other wall. This is analogous to saying “Saturn is at 20º Aries” from a sidereal reference. I am standing in one place, but in reference to one wall I am 3 feet away, and in reference to another wall I am 10 feet away. Similarly a planet is in one place in space, but in reference to a sidereal point it is described as 20ºAries and in reference to a tropical point it is described as 13º Taurus.
Now, suppose that someone throws me a ball. I will be able to catch that ball when it gets close enough to the same point in space where I stand. You can say, I will be able to catch it when it gets close to being 3 feet away from one of the walls, or you can say I will be able to catch it when it gets close to being 10 feet away from the other wall. The important part, though, is that there is only one ball and it only occupies one point in space, but we can describe that point from two different references, so on once hand we can say its at 3 feet and on the other hand we can say its at 10 feet. 3 feet from one of the walls is 10 feet from the other. The transiting planet is analogous to the ball moving towards me. I am analogous to a natal planet, or some other point in the natal chart. When the ball gets close enough to me, it becomes catchable. Similarly, when the transiting planet gets close enough to the transited point, it starts to have an effect. Whether you describe that point as 3 feet from one wall or 10 feet from another is irrelevant.
So, the zodiac system you use does not change the timing of transits.
It does change the idea that “Saturn is in Scorpio right now.” But that is not really very relevant because your own relationship to Scorpio also changes when the system changes from sidereal to tropical. What is important about transits is not really if Saturn is in Scorpio or Sagittarius right now, what is important is how Saturn’s current position relates to the various points in your chart.
Hopefully my example might make it clear?