We Don't Know
I was thinking about this yesterday as we know when Aegon's coronation was (I mean it's obvious the first day of the first month of 1 AC) so was wondering if we had any information on Robert's one.
I have tweeted Elio and Linda of westeros.org and they have come back saying we have no known date for the coronation.
@westerosorg Hello! Do we know the date of Robert's coronation? I've looked on the wiki and had a scan through the books but all I can find is late 283 AC.
No, that's one George hasn't given a precise date for.
What follows is speculation on my part for when the coronation could have happened.
A Song of Ice and Fire
The wiki page on Robert has this to say about his coronation.
In late 283 AC, Robert was crowned king.
It gives no extra information on the date. However, we can also deduce that it must have been late 283 AC from the following information.
Joffrey wishes to be crowned within a fortnight and so must believe that the preparations are possible for this to happen. He says it as if it's possible but not always the case so I'd say 2-4 weeks of preparation is probably normal.
Joffrey stood. His red satin cape was patterned in gold thread; fifty roaring lions to one side, fifty prancing stags to the other. "I command the council to make all the necessary arrangements for my coronation," the boy proclaimed. "I wish to be crowned within the fortnight. Today I shall accept oaths of fealty from my loyal councillors."
A Game of Thrones, Eddard XIV
Daeron Targaryen was crowned within the 2-4 week window as well after learning of his fathers death and that includes travelling to and from Dragonstone so it certainly seems possible.
His son and heir, Prince Daeron, departed Dragonstone within the fortnight after learning of his father's demise and was swiftly crowned by the High Septon in the Red Keep. He chose to be crowned with his father's crown—a decision likely intended to quell any remaining doubts about his legitimacy.
The World of Ice and Fire, The Targaryen Kings: Daeron II
The Sack of King's Landing was in 283 AC and was a race by the main rebellion forces of Baratheon/Stark/Arryn/Tully and the Lannister forces to who would get their first. The Lannister forces beat them there and were shortly followed by the main force. This force was led by Eddard Stark as Robert was injured and so we can assume that Robert needed extra time to get there. He would also need time to heal a bit as an injured king at his own coronation wouldn't be the best sight.
With all this information I'd speculate that Robert was crowned within 6-8 weeks of the Sack of King's Landing.
Game of Thrones
I won't repeat stuff here as most of the information is the same as the books including Joffrey wanting to be crowned within the fortnight.
Joffrey Baratheon: I command the council to make all necessary arrangements for my coronation. I wish to be crowned within the fortnight. Today I shall accept oaths of fealty from my loyal councillors.
Game of Thrones, Season 1 Episode 7, "You Win or You Die"
As the show has pushed back the timelines starting from Robert's Rebellion so they can keep the Dany sex scenes the date is slightly different, especially as the show uses AL not AC. As such his coronation must have happened in late 281 AL and that is what is on the wikia linked above.