Which child should be evaluated first among the following?

The child who presents with bilious emesis requires immediate evaluation due to the risk of a serious underlying condition such as intestinal obstruction or volvulus. Bilious emesis indicates that bile is present in the vomit, which suggests that the obstruction is distal to the entry point of bile into the digestive tract. This can lead to dehydration, electrolyte imbalance, and potentially life-threatening complications if not addressed promptly.

Early intervention is crucial in such cases to prevent worsening of the child's condition, as the presence of bile in vomit may rapidly indicate a surgical emergency that needs to be managed quickly to avoid complications such as bowel necrosis.

While the other scenarios present important issues requiring attention, the potential for an obstructive process with bilious emesis positions this child at higher risk for acute deterioration. The dislodged gastrostomy tube may require timely care but usually does not pose an immediate life threat. The vomiting after a handlebar injury raises concerns, but without immediate signs of distress, it can be assessed after the more urgent cases. Similarly, inconsolable crying in a very young infant may indicate distress or pain but does not provide the same immediate risk of acute medical issues that bilious emesis does.