An uncomplicated upper respiratory tract infection in a 12-year-old is managed on clinical grounds; it is a viral, self-resolving condition for which imaging offers nothing.
Two principles converge here — investigation appropriateness and radiation protection in children (ALARA). Radiology is reserved for situations that raise concern for lower respiratory tract disease such as pneumonia or respiratory distress, none of which is present. The correct next step is therefore to obtain no radiological investigation.
The remaining choices fail on appropriateness: a chest radiograph is the entry-level study only when pneumonia is suspected; HRCT is for chronic interstitial or airway disease and carries a high dose; and CECT layers contrast on top of an even larger radiation burden — all unjustified for a simple cold.
So no imaging is indicated.
\[\boxed{\text{No radiological investigation}}\]