It seems there is various versions of Yudhistir's coronation. One before the war. One after the war. Which of these is authentic and when did the coronation take place w.r.t war?
Yudhisthira became yuvraj of Hastinapur just before Lakshagrah incident.
After false news of Yudhisthira's death in Lakshagrah incident, Duryodhan became yuvraj of Hastinapur. But when Yudhisthira found alive in Draupadi svyamvar, he was appointed as the king of Indraprastha.
He became Samrat after Rajsuya yajna.
Later he lost his kingdom in dhyut-krida to Duryodhan and got 12+1 year exile. After Mahabharata war, he became the king of Hastinapur.
Later, he performed many yajna and became Samrat.
Please visit this research article for timelines.