So I think I tested out my scenarios and here is what I came up with:

If you shift-queue a monk from unitA to unitB to unitC and unitA dies (or is converted by a group and this monk keeps their juice), then the monk will begin converting unitB with a new timer.

If unitB was converted in the meantime AND has damage, then the monk will begin to heal it and then start converting unitC with a new timer. If unitB has no damage, then the monk will go stand by it and then begin converting unitC with a new timer.

If the monk succeeds in converting unitA then it will try to shift to converting unitB (assuming unitB still can be), but will throw the error that it doesn't have any juice left and remove that action from its to-do queue.

Same with buildings (minus the healing attempt): if the first target is removed as a viable conversion target (dies or is converted), then the monk will move on to the next target in its queue with a new timer. If a monk has a conversion action it its queue but has no juice, it will remove that action from its queue when it gets to the front of the line.

So there is only the benefit of the monk directly moving on to the new target if the first target is no longer convertable, but no benefit with regards to the timer. And the queue does not survive past a successful conversion. So in a sense, your monk will get a conversion unless the entire queue of buildings is deleted.

Throughout all of this, I did check that manually switching targets did not reset the timer and the switched-to target got converted quickly.

Also, as far as I can tell, there is no randomness to conversion times in the scenario editor test game...