When an ability says it interrupts, the actual implementation is just to apply a stun for the remainder of the round. So if you're faster, then you stun the opponent before they can perform their action for the round. If you're slower, then you stun them after they perform their action and the stun wears off at the end of the round, which in most cases accomplishes nothing.
A handful of multi-round abilities are actually interrupted by being stunned, but it depends on the multi-round ability's implementation: most of them actually aren't interrupted. Like you pointed out, Wish isn't interrupted: that's because Wish applies a buff that performs the actual healing, and being stunned doesn't remove the buff.* The same goes for Emerald Dream/Hibernate: these are buffs that put the pet to sleep and perform healing, and they aren't removed by being stunned. Pump/Wind-Up apply buffs that can be consumed later, so they aren't affected. The only abilities that are really interruptable are the ones that lock you into 2+ rounds of the same ability: things like Stampede/Swarm/Hunting Party, and Rampage/Mana Surge. These cases are pretty few and far between.
*Of course, if you're faster, then you can stun them before they can apply the Wish buff (or Emerald Dream, or Burrow/Lift-Off, or whatever). They can choose the same ability again the next turn, since the interrupt doesn't put the ability on cooldown any more. Effectively you've cost them a turn while doing a bit of damage--this makes interrupts perform a somewhat different function than in the regular game, but they're still pretty advantageous.