I am writing a short story on someone's journey from Basel to Hanover and was wondering if it was a reasonable amount of time to do so, if not I would love some other now day German city that would be a reasonable destination
I found a trip report for a journey between Berlin and Strasbourg in the year 1778 by an anonymous French correspondent who left Berlin on May 3 and arrived in Strasbourg on June 9. The traveler relied on stage coaches and similar conveyances for the entire trip.
Johann Bernoulli (ed.), Sammlung kurzer Reisebeschreibungen, Band 3, Berlin 1781, pp. 135-170: "Reise von Berlin nach Strasbourg. 1778."
This person took a leisurely trip and stopped (sometimes for days) at various places to contact notable people in the towns they were passing through, paid visits to various mines and factories, and did some touristic sightseeing as well. The route taken was: Berlin - Brandenburg - Magdeburg - Halberstadt - Wernigerode - Goslar - Clausthal - Göttingen - Kassel - Marburg - Frankfurt - Mannheim - Rheinzabern - Strasbourg. Extracting only the actual travel time, I counted a total of twelve days, seven days of which were needed to cover the distance between Göttingen and Strasbourg.
Given that travel from Mannheim to Strasbourg took two days, I would estimate that travel between Strasbourg and Basel would also require two days. Considering the trip between Göttingen and Hanover, I would estimate it to take 1½ days. This would result in a total travel duration of 10½ days for a trip from Basel to Hanover.
As ccprog pointed out in comments, late spring and earlier summer provide near optimal weather conditions for the route taken, in particular the hilly / mountainous region of Hesse between Göttingen and Frankfurt. At other times of the year progress may have been slower on that portion of the trip due to road conditions.
In the 18th century, long-distance travel by foot was a thing among people who could not afford to travel by coach. At the age of twenty, Johann Sebastian Bach famously walked from Arnstadt in Thuringia to Lübeck to see the famous organist and composer Dietrich Buxtehude. He did so in the latter half of October of 1705. It took him approximately two weeks to cover a distance of 250 to 280 miles (his exact route is not known), meaning he progressed at a rate of approximately 20 miles or 32 km per day.
Kerala J. Snyder, "To Lübeck in the Steps of J. S. Bach." The Musical Times, Vol. 127, No. 1726, Dec. 1986, pp. 672-677 (scan of first page)
While it seems unlikely that someone would attempt a trip from Basel to Hanover on foot, at a distance of 420 miles it would have taken three weeks for a person at the same speed and level of fitness as J. S. Bach, of whom it is known that he accomplished other long-distance trips on foot, such as a journey from Ohrdruf to Lüneburg (a walking distance of about 190 miles) at the age of fifteen. Again, adverse weather would have prolonged such a trip.
For tracing postal routes through Germany in the second half of the 18th century, I have found two maps with helpful notes on times:
Neue und vollständige Postkarte durch ganz Deutschland (1764) measures distances in "Posten", the normal distance one rider will cover before being relieved.
In Deutschland sind die Posten von vier, in Frankreich u. Niederland aber nur von zwey Stunden Weges.
In Germany, posts are a length of four hours, but in France and the Netherlands only two hours.
Post Karte Durch ganz Deutschland (1795) is a bit late for your purpose, but shows some later variants. It measures distances in German miles of 7.4km.
For the following routes from postal station to postal station distances are measured also in "Posten". I have tried to only take routes where a traveller would have had the opportunity to take a coach, with only some short variants where only the later map shows a stage coach.
One four-hour "Posten" roughly relates to 2 German miles. As I understand it, the speeds given relate to a courier on foot that would travel from one station to the next, and then rest. So if I give "traveling hours on foot" further down, this does not mean an average person could walk the whole distance at that speed, but the time a letter would take when transported in Estafette (dispatch).
Riders or stage coaches could achieve double that speed, provided road conditions were appropriate, and while also changing horses at each station. Especially in mountainous terrain, they would probably be slowed down significantly.
Left of the Rhine: Basel to Cologne
This route crosses into France at the city borders of Basel in Saint-Louis, travels along the river in Alsatia to Strasbourg and only reaches the German Border at Wissembourg. From there, it goes to Worms, leaves the river valley to reach the Hunsrückhöhenstraße to Koblenz. From there, it is straight traveling along the river to Cologne.
1fp is a French "post" of 2 hours,
1p is a German "post" of 4 hours. The total distance thus is 31.25 posts or 125 traveling hours on foot. In a coach, seven days would be reasonable.
Basel 1fp Saint-Louis 2fp Kembs 1.5fp Ottmarsheim 1.5fp Fessenheim 1.5fp
Biesheim 1.5fp Marckolsheim 1.5fp Friesenheim 1fp Krafft 2.5fp Strasbourg 1.5fp
Brumath 1.5fp Haguenau 2fp Surbourg 2fp Wissembourg 1fp (inclusive border crossing?)
Nieder-Otterbach 2p Landau 1.5p Speyer 1.5p Oggersheim 1.25p Worms 1.5p
Alzey 1.5p Kreuznach 1.5p Dörrebach 1.25p Simmern 1.5p Ehr (Halsenbach) 1.5p Koblenz 1p
Andernach 1.5p Remagen 1.25p Bonn 1.5p Köln
Right of the Rhine: Basel to Cologne
This route goes through Germny all the way. From Basel, it takes a slight detour away from the river to Freiburg and then goes along the base of the mountains to the east (Schwarzwald and Kaiserstuhl) via Rastatt to Heidelberg. From there, it follows the Bergstaße to Frankfurt. The Cölnische Hohe Heer- und Geleitstraße crosses the Taunus and Westerwald mountains to reach the Rhine only shortly south of Cologne. To enter the city, a traveler would have to cross the river with the Deutzer Gierponte.
The total distance is 32 posts or 128 traveling hours on foot. In a coach, seven days would be reasonable.
Basel 1p Kaltenherberg (?) 1p Mülheim 1.5p Freiburg 1.25p Emmendingen 0.75p
Kenzingen 1.5p Friesenheim 1p Offenburg 0.5p Appenweier 1p Bühl 1p Rastatt 1p
Grunwinkel (Karlsruhe) 0.5p Durlach 1p Bruchsal 1p Wiesloch 0.75p Heidelberg 1p
Weinheim 0.75p Heppenheim 2.5p Darmstadt 1.5p Frankfurt 1p
Königstein 1.5p Würges 1.5p Limburg 1p Wallmerod 1p Freilingen 1.25p
Giehleroth 1p Weyerbusch 1.25p Uckerath 1p Siegburg 1.5p Köln
On the Rhine
Could a traveller take a barge? Work on regulating the Rhine started in 1685, but the main work to reroute it in the Upper Rhine valley took most of the 19th century. Traders would run barges downriver even before these works, but might be forced to pause depending on draughts and floods. Since traveling by coach would take roughly the same time, and it would have been more reliable, there is not much incentive to take a ride on a barge.
The situation between Mainz and Cologne is different. On one hand, coaches are slowed down significantly because they have to cross mountain ranges, whatever side of the river is choosen. On the other hand, the river is shallow and has risky passages through the narrow Middle Rhine valley (think of the ominous Loreley rock). Is one day gained and the avoidance of roads so bad that you would have to leave the coach and walk on foot worth the risk of drowning to you?
Cologne to Hannover
The route starts on the left side of the Rhine. At Düsseldorf, another reaction ferry allowed a river crossing and going on to Duisburg. To trace the route from there through the Ruhr region is quite difficult. Industrialization has changed the landscape quite thoroughly, old proud towns were reduced to city districts, and former villages rose to major population centers due to the coal mines below them. The main postal route of the time seems to have taken a slight detour to the north to Münster before joining the older Hellweg to Paderborn. Another older map shows the route branching of at Hildesheim, but on the newer maps coaches take a direct route.
The total distance is 22 posts or 88 traveling hours on foot. In a coach, four days would be reasonable.
Köln 1p Dormagen 1p Düsseldorf 1.5p Duisburg 1p Brandenburg (Oberhausen?) 1p
Dorsten 1p Haltern 1p Dülmen 1.75p Münster 1.5p
Warendorf 1p Herzebrock 1p Neuenkirchen 1p Paderborn 1.75p Detmold 0.5p
Lemgo 1.5p Alverdissen 1p Hameln 1p Springe 1.5p Hannover