Lots of factors, depending. More can always be said regarding information dissemination in previous periods, so if anyone would like to address the topic, please don't let this post stop you!
For the meantime, since you mention That Norman Git, here's u/yomoxu and u/x--BANKS--x considering your example question.
As u/DanKensington already pointed out, it depends. Regarding the Roman Empire:
The transfer of information in general relied heavily on carriers, who mostly traveled probably by foot. Some kind of vehicles where only used when the message was important, e.g. horses (but horses where most of the time reserved to the military).
Using carriers also came with problems. Thanks to M. Tullius Cicero (106–43 BC) we know that some messages were just not delivered (cf. Cic. Fam. 2,10,1 and Cic. Fam. 2,13,1). Maybe the carriers were robbed or just unreliable. Additionally, the indifference of some carriers altered the time when a message would be recieved by the recipient (cf. Cic. Fam. 8,13(12),4 and Cic. Ad Qu. 3,1,23).
During the reign of the roman emperors (which began with Emperor Augustus, 63 BC–14 AD) an instituiton was established later called cursus publicus. It could be used by officials and carriers to deliver important messages.
The cursus publicus was an organised system. Roman places had to provide travellers with shelter and had to lease means of transportation to the travellers.
Now I would like to give some examples of how fast some information was transported.
S. Roscius (once defended by Cicero in 80 BC. The speach to this incident by Cicero: Pro Sexto Roscio Amerino) death was recieved by Chrysogonus (who wanted S. Roscius dead) near Volaterrae after four days. This is a distance of roughly 300 km, 186 miles.
The information that Nero (37–68 AD) was the new Emperor, was recieved after 35 days (from Rome, Italy, to Oxyrhynchus, Egypt. According to Google Maps that is a distance of 3789 km, 2354 miles, if travelled by foot) and that Traianus (53–117 AD) was the new Emperor after 28 days (from Rome, Italy, to Edfu, Egypt. According to Google Maps that is a distance of 4137 km, 2571 miles, if travelled by foot).
Bibliography:
Kolb, Anne, Transport und Nachrichtentransfer im Römischen Reich (Klio. Beiträge zur Alten Geschichte. Beihefte. Neue Folge, Bd. 2), Berlin 2000.
Riepl, Wolfang, Das Nachrichtenwesen des Altertums. Mit besonderer Rücksicht auf die Römer, Leipzig/Berlin 1913.
I hope this helps a little bit. I'm an undergraduate history student, but I tried my best. Due to the fact I only used german sources, this english one might be interesting (I have not read it, though): Ramsay, A.M., The Speed of the Roman Imperial Post, in: The Journal of Roman Studies 15 (1925), pp. 60–74.