NICARAGUAN SIGN LANGUAGE comes close to answering the initial question, closer than most. However....
1) There were toneladas of students enrolled at that school where they met and formed a (language) community. They were of different ages, some older and with more ideas and experiences than the younger younglings. NONE were infants, and ALL were well older than the 2-year old marker.
2) They were NOT enrolled at the Villa Libertad Vo-Tech School of Managua from birth. It was a school, not an orphanage.
3) It seems MOST had been using one or another system of hand-signals for simple actions & requirements (comida, agua, comer, beber, dormir) called "memeticas" in the Spanish of Nicaragua. How "common" these gestures are from household to household and from village to village is debatable, but apparently there was some kind of an input from the various students before contact with one another.
4) Initially, the "Lenguaje de Signos Nicaragüense" is more of a pidgin, a mixture of partial systems of communication through hand signals that developed into a "manu franca" for all the students to use. With time, it underwent creolization, which has been documented. Syntacic language, Here we come!!!
So.......Lenguaje de Signos Nicaragüense did not develop out of a complete vacuum. Various (incomplete) systems of hand-sign "proto-languages" were already in existence and made contributions via the student-interlocutors.
It's pretty close, closer than most other options, but still not the total "ex nihilo" from birth scenario in the initial parameters of the question set.
Pero todavia soy aficionado y fanatico de la historia de Lenguaje de Signos Nicaragüense !!!!!! ;^)
The situation is similar enough that we can be pretty sure that something very similar would happen if the things were changed to live up to the specifications given - it's seriously close enough for that.