Literalmente você pode traduzir sua frase como o seguinte : "The baby was chocking, but 'was unchoked' after receiving (emergency) treatment from the first-aid team/personnel." Mas ainda melhor : "... but his breathing was restored after receiving the Heimlich maneuver from the emergency medical services (EMS) team/personnel" ou "...but obstruction to his breathing was 'dislodged' by the emergency medical services (EMS) team/personnel" entre outras descrições parecidas.

não existe uma tradução exata para "desengasgar". seria tipo "to clear the throat" ou "to stop choking". então seria "the child was choking but stopped choking after the first-aid team attended him/her"