A mushroom cloud from space looks almost flat and barely above the ground, so all you see is a circular feature. Because the mushroom is vertical, you don't see the stalk unless it's very near the terminator, where it may not be visible at all.
This feature is mushroom shaped but sideways parallel to the ground - no explosion of any type can do that. It also follows ground features.
Also, I can't find much professional information about what a nuclear weapon would do on Mars, but it's really important to remember that mushroom clouds are an effect of Earth's thick atmosphere and gravity, both of which are much lower on Mars. High altitude tests (starting around 30 km, where the atmosphere is still thicker than Mars' "sea" level) do not normally form mushroom clouds, but spherical ones, like this one from Hardtack I Teak (altitude ~50 km, the closest test picture I can find to the atmospheric pressure at Mars ground level, which is around ~45 km):
I believe a nuclear explosion on Mars should more closely resemble a high altitude test such as this.
A bunch of other minor points in the video make no sense. A big one is the claim that, "NASA cut off the live feed" just before Sliding Spring's closest approach, however:
1. This video was not live
2. This video was not from NASA but the Indian Space Research Organization
3. The video was not even from Sliding Spring's encounter
4. The video was not cut off
5. For that matter, the video was not captured as a video, but was a series of still images assembled into a video for release to the media
The "cut off the live feed" claim accompanies every. ***. ****. One. Of these claims, and the only time it was even a theoretical possibility was
this one in which they actually delayed cutting a live feed to get what they hoped would be a stunning moonrise (but turned out to be more of a blurry gray turd).
Edit: More on mushroom clouds from space.
While trying to find a picture of a nuclear test from space, it occurs to me there's a reason none exist: EMP. Above-ground tests tend to fry satellites within line of sight, so pictures are taken by mechanical film cameras, digital is out of the question as is electronically controlled film cameras like most early spy satellites. To produce a cloud matching the size of the feature in the video would require something much larger than the Tsar Bomba, which took out a large number of satellites. The Indian satellite would not have survived.
However, other things create mushroom clouds, so here's some examples:
This picture, you can see the stalk, but this is a *much* lower altitude picture taken with a *much* higher resolution camera:
You still can see the dimensions of ground features, let alone the plume. Because this was taken so close, it's not at all comparable to the Mars picture.
Here is a higher altitude picture of a volcanic plume:
This was still taken at much lower altitude and higher resolution than the Mars images, but its at least going in the right direction. The farther you get above a planet, the closer all of its features become to two dimensional. By the time you get a substantial part of its face in frame, it's visually indistinguishable from a flat disk except very close to the limb. This picture is close to that point, though still not quite as close as the Mars pictures in the OP.
