Generally, Pilate didn't care what happened to the body of Jesus once it had been determined that He was dead. Pilate's concern was only that the sentence had been carried out. That being verified, he could give the order for the soldiers to hand the body over to anyone he wished. So, if a family member wanted to claim the body of a "criminal", they would approach the governor for permission to do so. Without specific instructions, a victim of crucifixion would be removed by the Roman soldiers on duty and hauled to a nearby valley called "Himnon" where it would be unceremoniously "dumped" into the pit located there with the rest of the city's trash, then sprinkled with sulphur powder and left to burn and rot.
It is unlikely that Mary was in a state of mind OR had resources to acquire the body of her Son, and the disciples had all fled the city in fear for their own lives.
But, to avoid further humiliation of his Lord, Joseph hastened to secure the body for proper Jewish burial lest the priests, hypocritically "concerned about the approaching Sabbath", encouraged the soldiers to immediately carry out the usual practice. They must have been upset when they discovered that Joseph had done so, so they hastened back to Pilate to request a guard be posted over the grave to prevent the disciples from "...stealing his body and proclaiming him 'raised from the dead', making the last deception worse than the first!" (Matt. 27:64)