Context is Important
First, you need to think about the time scale you are using. In my experience, most castings of sending do not happen "in combat and other fast-paced situations," and therefor, time is not even being marked by rounds or limited by actions. In this context, "immediately" simply means 'before the next event', whether that event is narrated by the DM or proposed by another character. Once the caster of sending begins their spell, they are allowed to complete it, state the wording, and hear the response, before anything else happens.
What if rounds are being used?
While uncommon, it is possible for a caster to use their action during a combat or other fast-paced situation to cast sending. In this case "immediately" takes on a more exacting definition. We can still use the plain English meaning of "immediately", but we need to apply that meaning to a situation in which time is measured in rounds and creatures are permitted to act in initiative order. In this case, both sending and receiving the communication occurs during the caster's action. The reply is received before any other creature can take their own action.
What about reactions?
Suppose another creature near the caster began the caster's turn with the readied action, "As soon as the caster begins casting, I do X". This creature's reaction is then resolved as "When the trigger occurs, you can...take your reaction right after the trigger finishes". Now it is up to the DM to decide what "right after the trigger finishes" means, ideally in a way that is consistent with their other rulings. A DM might rule that "right after the trigger" means the readied action occurs at the completion of the sending spell, that is, after the receipt of the immediate reply, but superseding the next action in initiative order.
However, the DM might rule that parsing the trigger to be the start of casting is acceptable, so that the readied action could actually take place before the completion of the spell itself. In that case, the readied action occurs before the immediate reply simply because the reaction interrupts the caster's action and the reply occurs only after the completion of the spell (while still within the caster's action).
Note that this second resolution would be similar to the case of a reaction such as Counterspell. If Counterspell was successfully cast as a reaction to the casting of Sending, the Sending would be interrupted and the immediate reply would not be received, since the outgoing communication was never sent.
Can 50 words be exchanged in 6 seconds?
While it may seem implausible that a 25 word communication can be sent, heard, a response composed, and 25 word reply sent all within one combat round, a careful reading of the Sending spell makes it clear that this is not spoken communication. From Sending (emphasis mine):
You send a short message of twenty-five words or less [sic] to a creature with which you are familiar. The creature hears the message in its mind, recognizes you as the sender if it knows you, and can answer in a like manner immediately. The spell enables creatures with Intelligence scores of at least 1 to understand the meaning of your message.
Note that the communication sent is not spoken, rather it is a mental feeling/image. There is no requirement that the recipient speak the same language as the sender, rather the spell itself provides the means of understanding. Thus, the sending and receiving is not expected to take the time needed to verbalize 50 words of spoken language. The 25 word restriction is understood as a limit on the complexity of the ideas expressed rather than a duration of time. There is nothing implausible about this happening 'immediately' (within the time of the caster's action).
Compare this with the cantrip Message:
You point your finger toward a creature within range and whisper a message. The target (and only the target) hears the message and can reply in a whisper that only you can hear.
Here, the message itself must be whispered. That is, the caster must physically speak the words within a specific language, subject to the limitations of how much content may be expressed in a certain amount of time. While the recipient is the only creature that hears the message, they hear it physically (not in their mind) and must understand the spoken language to comprehend it. Crucially, there is no reference here to an immediate response, so a DM is free to decide that messages of a certain length may take longer than one action to speak, and that the recipient's response may take place during their own action (rather than the caster's) if lengthy enough.