Disclaimer here, I'm on my phone and on vacation, but I'll try to do my best since this is in my wheelhouse! I should note first that whole books have been written on these subjects, but I'll try and do the abridged version.

The ancient Germanic peoples that we know of come principally from Roman sources. These sources name tribes like the Franks, Alemanni, etc, many of whom would go on to form the nations of France, Germany, etc. In this way, it is easy to see these peoples as distinct and unified groups, but this was not necessarily true for most of their history.

The grouping of Germanic peoples into distinct conglomeration (Franks, etc) was a Roman habit. Tribes were typically grouped not by culture, but by geography. Ergo, a Saxon was a Saxon to a Roman because he lived in the area designated by the Romans as Saxony. In reality, culture is much more fluid; there is no point where "Saxon" culture ends and "Frankish" culture begins. Instead, the two cultures are actually vastly similar, and the differences in cultures merge into each other since culture is a spectrum. There is no hard cut off. A second, and equally important point, is that as it turns out identity is more fluid than one would think. So, a Frank might consider himself a Frank as long as it is beneficial to him, but once that identity is no longer an asset (say, the Romans go to war with the Franks), he may change this identity.

This has some important implications. Most importantly, it means that Franks, and Germanics as a whole, did not view themselves as a singular cultural identity. When the Vandals (and friends) invaded Gaul in the 4th century, they did not do so as the Vandal nation, but as a tide of small Vandal groups. These groups would fight with each other just as much as they fought the Romans sometimes.

But to answer the main question: How did these disparate groups become nations? Well, there is a basic story for each tribe, but we can understand the general idea by looking at some case studies below:

Let's look at the Vandals. In the 4th century, the Rhine frontier collapsed and a whole host of Vandals, Alans and other barbarian groups crossed the frontier and pillaged Gaul. The Roman response to the catastrophe was slow, and so the invasion began as a somewhat disorganized ordeal made up of small bands of raiders. The barbarians were anything but stupid though, and knew that a response would eventually come, and that when it did, small raiding bands would have no chance. The best way therefore to survive was to collect these groups into a singular entity. And thus, a new Vandal horde was created. This Vandal horde was unique in that it took many of the bands that the Romans knew as Vandals, but also Alans and other groups, and forged them into a singular identity. This super group then went on to conquer Spain, and then North Africa.

In a similar way, we can look at the Franks. When Roman power in Gaul simply ceased to exist in the fifth century, Franks began slowly migrating and taking over land across the frontier. These too came in smaller groups, and initially their holdings we're dominated by small warlords. However, as time went on, stronger warlords began taking over others and exerting power over their rivals, thereby creating their own kingdoms. Eventually, this culminated for the Franks in Clovis, who essentially took all these groups who simply shared similar cultural shapes, and molded them into a singular identity, the Franks who eventually took France.

There is of course a lot more too it than just this, including economic and social motivators. Nevertheless, I hope that was at least helpful, and I implore you to take a look at some of the references below for further reading!

Sources: Empires and Barbarians: a fantastic read on Germanic identity (and other topics) in the closing years of the empire.

Warfare in Roman Europe 350-425: Discusses mostly how the Germanics viewed themselves and were viewed within the confines of the Roman army. Advances some interesting ideas/points.

Tacitus' Germania: Essentially the best source we have on Germanics of the middle empire period.

u/Jollydevil6 has already answered the part about the ancient Germanic people, I just wanted to add about the transformation into modern nations.

It is fair to say that the transformation of antic Germans to modern nations has been strongly pushed as a theory by 19th century nationalisms, who wanted to define nations as eternal continuous concepts. In reality, modern nations identities have much more to do with medieval and modern history and events than with their antic alleged ancestors. To address the examples that you take:

1. Modern time Germany's culture and frontiers have virtually nothing to do with the Saxons as an antic and early medieval political entity, and can be much more explained by the history of the HRE, and the competition between Austria and Prussia to the leadership of the German peoples. The fact that there are three German (Austria, Germany, and Switzerland) and two Dutch (Netherlands and Belgium) speaking countries in Europe, not ten or one, can be traced to much later than Antiquity. There are most definitely no specific divisions along these lines that were already visible in the Antiquity and that could predict the later development.

2. I am not as familiar with the history of medieval and modern Scandinavia as I am with Central and Western Europe, but what I can say is that the modern frontier between Denmark and Germany has nothing to do with an Antique divide between North and Central Germans, and more to do with later events.

I am not denying the importance of the Antiquity impact in general, of course. It is absolutely sure that all the European (and to a large extent the world) history is hugely influenced by the Roman Empire. And some cultural divides (particularly linguistic divides) survived through the ages and still have impacts, particularly when reinforced by natural geography. And of course divides can evolve through the times (Eastern Greek vs Western Latin roughly evolved into Orthodox vs Catholic, for instance). But in general, one shouldn't over estimate the continuity of antic peoples and their impact on modern polities.