Aside from specific effects like Counterspell, interrupting a 1 action spell isn't possible, they are too fast. Not even the Mage Slayer feat allows it. Since you can't interrupt attacks either, this seems fair enough.
More is the pity. D&D went from having spell take several seconds (measured in segments before) to most spells the somatic component is appearantly snapping your fingers, you don't have to actually have material components in hand, and the verbal components are "go!" or something.
Of course you can interrupt attacks, it is called Armor Class. ;-)
Seriously, though, there are ways to stop attacks from hitting (which is really, in essence, interrupting the attack). The Shield spell is a perfect example, although there are others. As a reaction you add +5 AC, after the hit even, making it miss. What else would you call that?
A lot of timing elements have been removed from D&D for simplicity, but like others I think they went overboard. We allow spells to be disrupted during casting, but it requires a readied action, a successful hit, and a failed concentration check. Fortunately, this works both in the players favor at times and against them at other times.
And why are spells so fast? I can understand the verbal only, maybe, and certain others. But if you play the material components must be in hand, that takes time, if the somatic components are complex in any way, that takes time, if the verbal components are more than a single word, it takes time. You put all that together and IMO spells absolutely should be long enough to stop while casting.
Finally, the "action" they take has nothing to do with the time required. Actions lack duration, as discussed at length in other threads.