By this I mean, was the flame of the vestal virgins put out when Constantine converted to Christianity? And, if so, how did this go down with the pagans who believed the city would fall of it went out? Or was it abandoned before Constantine?
The Vestal shrine was closed in 394 as part of the Theodosian reforms, over 80 years after Constantine legitimized Christianity. The flame was either extinguished, or at least allowed to go out as part of that.
While Constantine made the Christian religion official and legal, he didn't ban paganism. So the two co-existed for a long time side-by-side. There were a number of decades where one could attend mass at St Peter's Basilica in the morning, and offer votives at the Vestal temple in the afternoon.