Burn wound infection or cellulitis: This is the most likely diagnosis given the history of a burn, blister formation, and subsequent swelling. The lack of pain does not rule out infection, as some infections can present with minimal discomfort, especially in the early stages.

Contact dermatitis: An allergic reaction or irritation from the burn or its treatment could lead to swelling and blistering.
Burn wound edema: Swelling is a common response to burns due to increased vascular permeability and can occur without infection.
Superficial thrombophlebitis: Although less common, inflammation of the superficial veins could occur, especially if the burn was severe enough to cause significant tissue damage.

Gas gangrene: A rare but life-threatening condition caused by Clostridium perfringens infection, which can occur after a burn. It is characterized by rapid progression of swelling, pain, and crepitus (a grating sensation or sound from the fragments of a fractured bone).
Necrotizing fasciitis: A severe infection that destroys tissue under the skin, including fat and the tissue covering the muscles (fascia). It's a medical emergency and can be fatal if not treated promptly.
Sepsis: Although the child does not have pain, the possibility of sepsis, a life-threatening condition that arises when the body's response to infection causes injury to its own tissues and organs, cannot be overlooked.

Erythema multiforme: A skin condition characterized by target lesions, which can be triggered by burns among other causes. It's less likely but could explain the swelling and blistering.
Toxic epidermal necrolysis (TEN): A severe skin condition usually triggered by medications but can be caused by burns. It leads to widespread skin necrosis and detachment, which is a medical emergency.