What happens in a civil case if the plaintiff doesn’t answer interrogatories in time?

It's hard to say on the interrogatories; it all depends on which state you're in.  In MA, they have 45 days to answer, then you have to send a final notice and give them an additional 30 days, then you can file for a default judgment.  You should check your state's Rules of Civil Procedure.

Of course, they did finally answer them, so it's more likely than not that the judge will let it slide, even though they were a little late.

The admissions, however, are another issue altogether.  Admissions need to be answered by the deadline and there is no extension (again, check your state's rules).  If they really did fail to answer admissions on time (make sure they did not file a motion to extend the deadline), then everything in the admissions is admitted as true and is established as a hard fact for the purposes of your case.

You should file a Motion to Strike their admission responses based on the fact they were filed late.  Again, it depends on who the judge is; he might side with you, he might accept their late answers and deny your motion.  Judges are usually stricter when it comes to admissions, but they're also not wild about deciding a case on a technicality.

Either way, this doesn't resolve the case.  Best case scenario: the allegations in your admissions are held as true, but you still will need to either file a dispositive motion or go to trial.