Quick little video example of mystery delay in arena:

https://youtu.be/Q4fmxq2YMQg

It's not the worst that's ever happened to me, but that is still really far from ideal. In that time, if Temari had been CC'd or killed, I would have gotten System Error (See Shereif's response for more detailed information on that).

Is it affecting only certain ninjas? No, this can happen to any ninja's mystery that is used after the first chase of the round. In the video, you can see that Naruto's mystery activates fine, but Temari's, and possibly Lightning Main's, mysteries were delayed.

Is it affecting only certain skills? No, this happens to non-prompt ninjas as well (I.E. Gaara barrier mystery queued, but he'll still do his standard attack that round).

When are you experiencing this problem? It happens more regularly when there's a change in chakra (I.E chakra gain, as in the video), or if a mystery is used near the end of the round (appx. 2 standard attacks remaining, not counting clones). This is thoroughly frustrating when using CC based lineups against a position 4 healer, for example.

Is it happening during specific in-game timed events or is it random? For the most part, this occurs in scenarios with more than 1 person involved. This isn't limited to just the timed events (Sage World Battlefield, Strongest Ninja, Matsuri, etc). Systems such as Arena (see video above) and even dueling against group mates have this issue as well. However, that's not to say that Survival Trial/Instances/etc don't delay, they just appear to less severely or frequently.