If the speed gap between the overtaking vehicle and the one being overtaken is minimal, will the overtaking distance be shorter?

The overtaking distance is the section of road that you travel from the start of the manoeuvre until you safely return to your lane. With a small difference in speed, overtaking takes a long time, and during this time you are driving at high speed, so you cover a long section of road. So a small difference in speed makes the overtaking distance longer, not shorter. The answer is: no.

For example: when you drive at 90 km/h on a motorway and overtake a vehicle combination travelling at 80 km/h, you need more than ten seconds and several hundred metres; at a speed of 120 km/h the manoeuvre is several times shorter. On a two-way road a long overtaking distance means that a vehicle coming from the opposite direction must be very far away. The Road Traffic Act requires you, before overtaking, to make sure that you have adequate visibility and enough space to perform the manoeuvre without obstructing anyone.

Legal basis: Art. 24 para. 1 point 1 of the Road Traffic Act (Prawo o ruchu drogowym)