Why do they decide this? There has to be some sort of impetus for it, and it would have to be drastic. Even if the Constitution is rejected, an attempt to somewhat revise the Articles was the main (less radical) alternative. If, as
@Napoleonrules writes, the Constitution would be rejected, revision of the articles would be the next option on the table, and practically everyone supported at least that.
The more unlikely course would be to not revise the Articles at all, which has the best chance of leading to disintegration of the Union (or rather: the Confederation, at that stage) due to hard-to-resolve conflicts between various states and the lack of a central authority powerful enough to succesfully mediate or adjudicate. But even then, a simple POD of "no changes to the Articles" would need a pretty impressive reason for that to happen, since revision was far more likely, and it's still not a fail-safe guarantee for later disintegration of the confederal US.
Let's take a step back and consider the kind of POD that could lead to disintegration. My best guess would be: George Washington is killed, someone else assumes command of the military, the war drags on longer, and somehow this (preumably less competent and less honourable) military leader attempts to seize political power at the end of the war. This military coup (or quasi-coup) is ultimately foiled or rolled back, but many people are now wary of centralised (military) authority. Federalism still takes off, but more as a local drive to further integrate the northeastern states. Strong nationalists like Hamilton are looked upon with suspicion, especially when they also have a military background.
The Articles basically last, but the confederation becomes ever more divided. Clearly, certain issues cannot be resolved adequately. Federalism still blooms in New York and New England, leading to an informal regional sub-confederation. Fervent attempts from these part to get the Articles amended lead to hostility from the south. As politics become increasinly embittered, and the Congress of the confederation does nothing but squabble anymore, New England eventually opts to secede and form its own, explicitly
federal government. New York, New Jersey and Pennsylvania also secede, but do not join New England. They form a new confederation of their own, along the lines of earlier proposals to revise the Articles.
The remaining states - basically everything south of the Mason-Dixon line - remain under the original Articles. I don't see them splitting apart, since they were basically in political agreement. Well... maybe if Virginia and North Carolina clash over competing border claims? I find that somewhat unlikely, but if it happens, you get two confederations in the south: Virginia-Maryland-Delaware and North Carolina-South Carolina-Georgia.
None of these would, in such a tumultuous time, really become involved with European wars. The New England federation and the New York-New Jersey-Pennsylvania confederation would both tend towards friendly relations with Britain, for purely economic reasons if nothing else. The deep south will just want to trade and not have that trade be taxed, and since the original Articles (under which all southern states would presumably remain) do not provide for a clear political executive figurehead... the man pushing for alliance with France - Jefferson - will have less opportunity to gain a paramount position from which he can effect that. (He should be happy nonetheless, since everything south of the Mason-Dixon line will look just as he imagined it should look. Especially if the deep south splits off. The upper south will then essentially be a Virginia-led confederation, exactly in line with his politics.)
On a more global scale, Britain would be jeering about the fact that the United States imploded within a matter of years, but the fact that a variety of republics would still have emerged from the process, alive and well, would probably give repulicans all over the world reason to feel vindicated.
Long term predictions: the more northern polities will have an economic advantage on the outset, although in OTL, there was a tendency to tax southern trade and use (most of) the profits to subsidise northern industry. This basically kept the south agrarian and made the north ever more industrial. Separated nations means the north cannot exploit southern trade like that. If the south can get wise and shake off its policies of anti-industrialism, they can use the profits of their own trade to build their own industry. This would make them a bit less dependent on that one perfidious institution, slavery. (They still won't abolish it, but it did get increasingly central to southern economy as the decades rolled by. If they can industrialise sufficiently before the 1840s, slavery will in any case be less central to their economy. By that point the British will presumably be stamping out slavery everywhere. Since slavery would be less developed in this ATL, there would be considerably fewer slaves in this TL's southern confederation(s). That would make abolishing the practice somewhat more realistic, the optimist in me dares to hope. But the pessimist in me suspects that, at the very least, the deep south would stick to slavery at all costs. In a confederal setting, each state decides on abolition on its own, so the deep south states can basically just refuse to abolish for all time. In that case, international outrage will eventually lead to a trade boycott, and it all ends very much like South Africa.)