I know from playing with long nosed spoons that they are not very useful when trying to get out of small, deep, grass walled depressions.......my own sense of golf forensics suggest to me that escape from this variety of bunker wasn't really possible until the advent of iron utility clubs.
Using the Old Course as an example and assuming cowering sheep weren't specialized to stacking sod....I would guess that the smaller stacked sod bunkers first began to appear maybe around the 1870's, initially as an erosion control technique, with its use gradually expanded as a maintenance method.... all the while receiving a favorable nod for its aesthetic appearance by club members....They became deeper as new club design allowed for the ability to escape from them.
This is just a designers query when trying to relate the equipment of the day to the design features a player would encounter.......and establish a period for both.
Can someone provide more definitive information?