Several memoirs of the period suggest that the Berlin to Vienna journey very likely could be completed in 12 days or less. This matches up fairly closely to @Eugene's estimate of two weeks.
However, one account suggests that someone with more limited resources and unexpected delays could easily take much more time.
The route they [1,2,3,4] usually seem to take is the following with locations mentioned in some accounts in [ ]:
- Potsdam (where Frederick II held audience with diplomats) or Berlin (where diplomatic corps was based)
- [Leipsic (Leipzig)]
- [Peterswald[a] (Petrovice) - cross into Austria]
Vienna to Prague Six days
William Wraxall (Wikipedia) claims to have left Dresden 24 November by carriage (he doesn't say how many horses), and arrived in Vienna on 30 November (probably 450-500km). “On my journey to Bohemia, Moravia, and Upper Austria, I only stopped to change horses” [1:297] He also claims that of this, he made the stretch between Dresden to Prague (probably 150-175km) in 38 hours [1:296].
However, this leaves us with the stretch between Dresden and Berlin, which is probably 180-200km directly north, but a number of accounts seem to suggest that rather than direct, they travelled via Leipzig (perhaps 275-300km)
Berlin to Leipzig Three Days
A number of travelers going from Berlin to Vienna seem to pass through Leipzig rather than go directly south. This does not mean that diplomats didn't take a more direct and faster route but gives us an outer estimate. Joseph Marshall (Joseph Hill) writes of a departure from Berlin on 1 June, and arrival in Leipzig on 3 June. [3:287] He goes on via Meissen to Dresden, but not clear how long it took since he appears to have stayed in Leipzig and Meissen for a time. Warning: This account may be, as many travel books of this kind in earlier times, a compilation of other circulating accounts posing as a single narrative and the contents should be treated with care. See this H-Habsburg entry.
Even if the diplomat traveled the "long way" via Leipzig, and the Leipzig to Dresden portion took the same amount of time as the longer Leipzig to Berlin portion, this still yields 6+3+3=12 days. If diplomats took a much more direct road from Dresden to Berlin, this time would likely be shorter.
On the Other Hand
Frederick II, who strictly limited interaction with diplomats who he kept in Berlin while holding special interviews in Potsdam, seems to have treated his own diplomatic service as a form of "covert taxation of the nobility" [5:100] which lead to the (claimed) impoverishment of some of his envoys, including Graf Jakob Friedrich von Rohde in Vienna.
If you didn't have good resources to pay off all the post expenses and "extra-post" expenses and other challenges and have carriages with fast horses, the trip from Vienna to Berlin could take much longer. One 1773 anonymous account reports that the Prague to Berlin segment alone took 12 days due to delays.[2:179-181]
Calculations of stage coach travel distances, etc. around the world are very useful for determining the outer bounds of what is physically possible. However, to the degree possible, we should try to answer questions like these based on historical sources when these are available. Some of course, may yield less plausible claims:
There is a claim of the fast trip of a courier from Berlin to Vienna in an alleged 48 hours in 1783:
From Vienna, where recruits are raising, and preparations for an approaching war are carrying on, with the utmost activity, and where the workmen are busy night and day in the arsenals. Add to all this, that a courier arrived at that city lately from Berlin, who had performed the journey of 144 leagues in 48 hours; the importance of whose dispatches was evident from the bearer's having nearly sacrificed his life to deliver them speedily to his Imperial Majesty, and from the orders immediately sent to all the troops in Upper Austria, as well as those in Hungary and Bohemia, to hold thesmelves in readiness to march on the first notice.[6:167]
It does not hurt to be skeptical of a claim like this and might be good to compare with similar claims in contemporary accounts of how far horses could travel in urgent situations. @Eugene's calculations suggests that this number is highly unlikely.
Sources cited as [Source Number:Page Number]
1800 William Wraxall, Memoirs of the courts of Berlin, Dresden, Warsaw, and Vienna, in the years 1777, 1778, and 1779. Archive.org
1774 The Annual Register, or a View of the history, politicks and literature of the year Gbooks
1773 Travels through Holland, Flanders, Germany, Denmark, Sweden, Lapland, Russia, the Ukraine, and Poland, in the years 1768, 1769, and 1770 Gbooks
1779 A View of Society and Manners in France, Switzerland, and Germany: With Anecdotes Relating to Some Eminent Characters. By a Gentleman, who Resided Several Years in Those Countries, Volume 2 Gbooks
H. M. Scott The Emergence of the Eastern Powers, 1756-1775 Gbooks
1783 The Gentleman's Magazine: And Historical Chronicle. Volume III, Gbooks