[JPS translation and verse numbering throughout, unless otherwise noted]
Chapters 1 and 2 - Locust
Chapters 1 and 2 describe a plague of locust:
What the locust swarm has left the great locusts have eaten;
the great locusts have left the young locusts have eaten;
the young locusts have left other locusts have eaten (1:4).
(NIV translation here)
Toward the end of chapter 2, these locust get expelled from the land:
But I will remove far off from you the northern one, and will drive him into a land barren and desolate, with his face toward the eastern sea, and his hinder part toward the western sea; that his foulness may come up, and his ill savour may come up (2:20).
Prior to that expulsion, the locust are personified as an enemy on horseback and God is described as leading the locust into battle against Israel:
The appearance of them is as the appearance of horses; and as horsemen, so do they run.
Like the noise of chariots, on the tops of the mountains do they leap, like the noise of a flame of fire that devoureth the stubble, as a mighty people set in battle array (2:4, 5).
The question at hand:
Should we read the first two chapters of Joel as allegorical
descriptions of the nations mentioned in the last chapter?
Short answer: no.
Here's a good explanation why not (From Marco Treves, "The Date of Joel" in Vetus Testamentum, April 1957):
The locust are compared to lions (1:6), horses (2:4), warriors
(2:5,7) and thieves (2:9). It would be very strange if they were
allegorical, for we should have similes within a simile, which would
be awkward and confusing and would defeat the very purposes of
allegory. Moreover, the prophet tells us that he saw the calamity with
his own eyes (1:16).
Personally, I see no reason not to take this prophecy describing a plague of locust both literally and seriously. If you don't think locust are a big deal, you probably never had a swarm complete ravage your crop. See:
Chapters 3 and 4
Chapter 3 starts with: "And it shall come to pass afterward...." This transition doesn't say much about the relationship between the two parts of the book.
Unlike chapters 1 and 2, chapters 3 and 4 are almost definitely describing real warfare between real nations. We have a description of “The day of Lord” (3:4) which involves a lot of nations coming to Jerusalem (4:2), God punishing these nations (4:4), and then some big war in the end:
Proclaim ye this among the nations, prepare war; stir up the mighty
men; let all the men of war draw near, let them come up. Beat your
plowshares into swords, and your pruning-hooks into spears; let the
weak say: 'I am strong.' Haste ye, and come, all ye nations round
about, and gather yourselves together; thither cause Thy mighty ones
to come down, O LORD! Let the nations be stirred up, and come up to
the valley of Jehoshaphat; for there will I sit to judge all the
nations round about. Put ye in the sickle, for the harvest is ripe;
come, tread ye, for the winepress is full, the vats overflow; for
their wickedness is great...And the LORD shall roar from Zion, and
utter His voice from Jerusalem, and the heavens and the earth shall
shake; but the LORD will be a refuge unto His people, and a stronghold
to the children of Israel (4:9-16).
Days of the Lord
If it's multiple calamities, must we infer multiple "Days of the Lord"?
We can. "Days of the Lord" in Joel 2:1, 3:3, and 4:14, need not be understood as a single eschatological moment, but rather any time that God's judgement is manifest in history. See for example Amos 5:18 who is talking to the northern kingdom of Israel at a completely different time period:
Woe unto you that desire the day of the LORD! Wherefore would ye have
the day of the LORD? It is darkness, and not light.
To understand the thematic and theological significance of these two prophecies and how they relate to other stories in Tanakh, it's necessary to figure out when this prophecy was given. For that question, see here.