Could the speed in which the plate is moving affect how quickly mountains rise?
The height of the Himalayas

Like Keith McClary says in his answer, there really are two factors in creating growing/shrinking mountains.


Mountains grow due to various reasons. In the case of the Himalayas it's like you must've seen in the documentary: the collision between the Indian and Eurasian continental plates. The material has to go somewhere and thus goes up (simplifying the geology a bit; I think that concepts like buoyancy these days also give some nuance to the subject, but the general picture stays the same). In the Himalayas, the two plates have moved into each other for at least 2500 km, so that's a lot of ground to move!





If that was the only process at play, there would be way higher mountains. I couldn't find a corresponding figure for the Himalayas, so consider instead the following present-day profile through Switzerland, running roughly north (left) to south (right).

You can see that the present-day Alps are at maximum 4 km high. Compare that against a reconstruction of the geology that should be present, running along basically the same profile:



You can see that the mountains could be as much as 20 km high! You can appreciate just how much mountain has disappeared, for another very young mountain range! So, the reason that mountains aren't as big as they could/should have been, is that they are cut down by erosion. Rain, snow, wind, temperature, all break down mountainous rocks into smaller broken pieces. Those pieces are transported away from the mountains as sediment in rivers (such as the Yangtze for the Himalayas).

So, the height of the Himalayas is very much controlled by the relative size of the two effects: how fast does the mountain range grow, and how fast does erosion break it down again. If the two are in equilibrium, the mountain doesn't shrink or grow. If erosion dominates, the mountain will slowly disappear over time. But in the case of the Himalayas, the mountain building generally outpaced the erosion, leading to the unique heights of the Himalayas. In this case, the speed of the collision is definitely the main factor!

The size of the Tibetan Plateau



Now if your question was also about why the Tibetan Plateau is so wide (rather than the Himalayas so high), I recommend you read https://oak.ucc.nau.edu/wittke/Tibet/Plateau.html , a very well-written summary of the three main hypotheses:


The speed of the collision has lead to a very step-wise straight faults toppling over each other, rather than folding over, which could lead to very long faulted sections. "Crust is thickened by the faulting and subsequent movement of large masses of rock, which are stacked one on top of another like cordwood."
When India hit the Eurasian plate it still had a heavy oceanic plate that was going under ('subducting') under the Eurasian plate. Now the oceanic plate has disappeared, and now the continental plate is being pulled underneath the Eurasian plate. "This process is reminiscent of taking a block of ice and pushing it beneath another ice slab, causing the latter to rise upwards. However, it is difficult to imagine how the buoyant Indian crust could be kept deep enough to get far beneath the plateau before bobbing to the surface. Perhaps the great speed at which India is colliding to Eurasia allowed this to happen."
The last hypothesis is simply that the continental collision puts a lot of heat below the plateau. "Like a hot-air balloon, the heated crust is buoyant and rises with the addition of light granitic material at the bottom of the Eurasian crust increasing the height of the Plateau."


This picture shows all those three hypotheses. 1=distributed shortening, 2=underthrusting, 3=lower crustal flow. In each case, a plateau is formed relatively far away from the actual collision between the two plates.


It is likely that, really, all three hypotheses together are at play. As explained on the website, at least the first two hypotheses are thought to be related to the speed of the collision.

So, yes, both the height and the width of the Himalayas are thought to be linked to the speed of the collision/relative plate movement!