The Dutch speak Dutch / the Deutsch speak Deutsch
As the famous quip goes, a language is a dialect with an army and a navy.
The historical linguistic reality is that south of the north Germanic languages and east of the Anglo-Frisian languages there used to be two Germanic dialect continua, conventionally called Low German (Plattdeutsch, German of the Flat Lands) and High German (Hochdeutsch, German of the High Lands). Like this:
The Uerdingen line (the Ik/Ich isogloss), one of the isoglosses separating Low German (to the north and west) and High German (to the east and south). Map by Slomox, available on Wikimedia under the CC BY-SA 3.0 Unported license.
Low German and High German differ principally in that High German dialects participated to the second Germanic consonant shift, which is the reason why in Modern German "ship" is "Schiff", "apple" is "Apfel", "out" is "aus", "two" is "Zwei", "wife" is "Weib" and "day" is "Tag". (The corresponding Dutch words, as a Low German standardized dialect, are much more similar to the English words: "schip", "appel", "uit", "twee", "wijf" and "dag".)
Berlin is firmly in Low German territory: so how come that today Berliners speak a High German variety and can no longer understand their Dutch Low German close relatives?
The answer is complicated, like most answers having to do with European history. The basic reasons are:
The territories which make up modern Germany and Austria were for a long time part of the Holy Roman Empire; politically the HRE did not account for much, but the imperial bureaucracy eventually standardized on a common bureaucratese called Sächsische Kanzleisprache, Saxon Chancellery Language, a form of High German which avoids the more extreme characteristics which would have made it utterly incomprehensible and alien to Low German speakers.
In time, people first became accustomed to the simple fact of life that official documents used a language similar but different from their daily speech. And then, when modern times came and education became free and compulsory, guess what common language was taught?
The territories which make up modern Germany and Austria (called "the Germanies" in the Early Modern period) have always been quite closely united culturally, whereas the Low Countries were always outside this cultural community.
And finally, around 1870 the nascent German Empire decided administratively that everybody should learn High German in school.
Yes, the famous translation of the Bible by Luther played a role, but it was a modest role. Luther's translation uses a central variety, neither too Low nor too High; it's main importance is that is strengthened the cultural links between the Germanies.
A simple small change in history
Coming back to the question, what change in history would make the Dutch speak German, the answer is that in a sense they actually do. Continental west Germanic has two standardized forms, one called Dutch ("Nederlands") and one called German ("Deutsch"); Dutch a standardized Low German dialect, and German is a standardized High German dialect.
While I cannot imagine a "simple" change which would have resulted in only one standardized form of continental west Germanic, I can easily imagine a relatively simple change which would have preserved the Low Germanicity of Saxony and Brandenburg and Prussia and Pomerania: Gustavus Adolphus does not die in the battle of Lützen; instead, he continues to lead the northern Protestant side in the Thirty Years' War and establishes the Corpus Evangelicorum as a polity separate from the mostly Catholic Empire. A political boundary between the northern and western Low Germanic dialects and the southern and eastern High Germanic dialects would have preserved the linguistic separation, which would have, in time, resulted in two standard languages.
And who knows? Maybe the Dutch would have adopted the Low German standard of their Deutsch neighbors...