According to the official script for the episode the Mountain feels "no undead camaraderie with the creature whatever". According to the script he steps to her side; no mention is made of hesitating so it looks as if he moved as quickly as he could.
He no longer wears his winter clothing, only scraps of cloth that reveal the rotting mess of his body, and the holes where the Hound slammed him down on Drogon’s spikes. He has a metal collar around his neck...
... but we don’t have a chance to properly register this as he’s making for Cersei, snarling and gnashing.
Feeling no undead camaraderie with the creature whatever, the Mountain draws his sword and steps to Cersei’s side, ready to slice the thing in two.
The creature is almost upon Cersei. It reaches for her --
-- and is roughly jerked to the ground.
Game of Thrones, Season 7 Episode 7, "The Dragon and the Wolf"
We don't know but it actually appears to be a lack of time for the Mountain to get in the way. The Mountain is a slow moving beast as we've seen previously but he does protect Cersei as his number one goal. We see him move in front of her for protection a few times, even in the same scene when Sandor approaches.
As for why he doesn't in this occasion he doesn't have enough time to actually block the wights path as it approaches quickly. However, if you watch closely in the background he does grab his sword but we don't see if he pulls it out or not. He also appears to be as startled as the rest of them when it starts approaching her.
Also note that once Sandor pulls the wight back both Jaime and the Mountain have moved into protective positions around Cersei. The Mountain has also drawn his sword at this point. This backs up the theory that they were just slow to react and were startled at first.
As for why the other 6 Queensguard, Jaime and Euron don't react:
- 6 Queensguard: Are startled and don't have the time to get there anyway and also probably thought The Mountain would handle it.
- Jaime - Too startled and again probably thought The Mountain had it covered.
- Euron: Doesn't really care but is resolved enough to keep it together so probably thought someone else would deal with it.
Out of Universe
It is to show Cersei that the threat is real and to scare her into making her "decision" to help them on her conditions. If you have The Mountain protecting her and stood in the way you wouldn't really get that.