For instance, was the Roman citizen informed of the military conquests of Gaul or an English citizen the results of the Battle of Hastings? Was there some sort of newspaper? Did they even care?
In your question, you specifically cite the conquest of Gaul, which is a great example of not only how news travelled by "channel" but also the impact it had.
Julius Caeser himself was responsible for the majority of the news being sent back to Rome, something he orchestrated brilliantly, by writing his account of the Gallic Wars in realtime (which can be read to this day). This wasnt what we would describe as a traditional autobiography, i.e. written years after events and with the purpose of securing someones legacy, but instead as news and propoganda to be used in at the time of its writing. Caeser knew the power of this, and would have his accounts sent regularly throughout his campaigns to Rome to be read in the Forum in public, specifically writing his accounts in the third person to make it more appropriate for this format. Obviously he did this to secure his position in the Roman Senate but also crucially with the popularii, the common people, by providing them with free news of his heroics in distant lands.
A consequence of invading a giant territory like modern day France was the immense wealth and the number of slaves Caeser also brought back, which acted as a constant reminder of his successes and would travel through large swathes of the republic before ending up in Rome, providing news and, crucially, evidence of what was happening. Finally if you add in the traditional avenues of news, such as trade, diplomats, bureaucrats, military leave, personal communication to love ones from military personnel and, again, propoganda through art etc, the network was extremely well connected.
Using art to promote communication was especially prevalent not only in the classical era, which docummented news and distributed it throughout nation states, but also in the 'dark ages' e.g. saxons, angels, mutes, Vikings etc. As these peoples didnt have a strong history in literacy, bards and troubadours would be commissioned to travel around kingdoms spreading tales of their patrons strength, wealth, piety etc as a form of promoting their brands through the guise of news (fake news anyone?). This would have been extremely powerful within these ages where formalised communication in writing and through rigid systems/structures was decaying.
Using the Roman example further, something which was crucial to news in ancient and medieval times was rumours and gossip. The word trivia comes from the Latin tres viae, meaning three roads. Roads allowed travel from different, far flung populated areas. Where these roads met would become focal points for the exchange of news from regions, whether with that being the sole purpose or a secondary one (e.g. trade).
Another example can be found during the height of the Mongolian Empire. The Mongolian empire spanned thousands of miles across land, ranging from (at times) Russian to Korea. Stations were set up every few miles with multiple horse which would be swapped out as riders with messages approached them, allowing for the fastest method of communication in human history until the 19th century. Granted whilst a large portion of the traffic was imperial and therefore between rulers and governmental forces, you cannot downplay the power of gossip and indirect communications mentioned earlier. These small hubs were beacons of knowledge and news, populated by a few individuals who would work there and, outside of work, go to their homes and villages to spread what they had heard.