"Immediately" probably means "within 1 round."
I have nothing to back this up except a couple of inferences and either a weak absence of evidence argument or Occam's Razor. That said, I still think it's the best we got until another jurist wows the court.
First, how does sending work? It doesn't say that the caster has a round to think of the message, nor does it say that it takes a round to transmit. A person who didn't read the duration could reasonably assume that the message is sent when the spell is cast (if you had a round to think of a message, why would the rules go to the trouble of duplicating the effect of Readying the Cast a Spell action?), and that transmission is Instantaneous. This is the reading that advantages the caster, certainly; if it took 1 round for messages to arrive, PCs would be sending potentially outdated information.
Is there a different reason, then, that sending has such a long duration? I think it's because a creature doesn't act outside their turn (unless specified), while communication is a flourish you may take on your turn (see "Other Activity on Your Turn," PHB p. 190). It seems reasonable that the recipient would answer on their turn.
Now it's quite possible that sending creates a specific rule about communication that overrides the general. But then we're back to not knowing why the spell lasts a round. If "immediately" meant "instantaneously" then sending would be Instantaneous. So I'm inclined to think "1 round" is the limit on "immediately." This does conflict with message not requiring immediate response, but I still think it's the best explanation, and consistent with sending's obvious place as "message but better."
Also, for what it's worth, asking yourself a question and then saying "uh" for 6 seconds does still feel like it is immediate under the "you know it when you see it"/"in human terms" test that Dale M posits in his answer.
