The following article gives an answer to this question not only when but also by whom:
It says it was about the time of the birth of Bach that German keyboard players adopted the system of a grand staff influenced by French publications.
I wonder where we would be today interpreting Bach and how Bach would have written his work if the full score and grand staff wouldn't have been developed at his time.
Only in the late seventeenth century, under the influence of French publications, did German
keyboard players adopt the system of keyboard notation on pairs of five-line staves. Pedal parts
were normally included on the lower staff, not on a separate one of their own as in modern organ
scores. Some manuscripts, especially those containing Italian or English music, still employed
the six- and eight-line staves used in those countries. In such manuscripts the placement of notes
on the upper or lower staff is an indication to use the right or left hand, respectively; a melodic
line exchanged between the two hands, as in the inner voices of many contrapuntal pieces,
wanders between staves. But many German musicians continued to employ the older form of
notation known as tablature, using letters instead of notes. This more abstract notation requires
the player to determine which notes belong to which hand (or to the pedals). Although more
economical—a composition could be copied using a half or a third as much paper—tablature is
harder to read and by 1700 was going out of fashion. J. S. Bach continued to use it well into the
eighteenth century, but only to notate brief sketches or when forced by lack of space to write a
few measures of music in the margins of a page otherwise employing staff notation.
One of the few substantial German publications of keyboard music from the first half of the century, the Tabulatura nova (Hamburg, 1624) of Samuel Scheidt (1587–1654), was printed
in open score, using a separate staff for each contrapuntal line; Scheidt nevertheless expected
players to transcribe this into tablature notation.
Looking for more information I’ve found this source:
The treatise Musica enchiriadis (AD 900) uses Daseian notation for indicating specific pitches, but the modern use of staff lines is attributed to Guido d'Arezzo (AD 990–1050), whose four-line staff is still used (though without the red and yellow coloring he recommended) in Gregorian chant publications today. Five-line staves appeared in Italy in the 13th century and it was promoted by Ugolino da Forlì; staves with four, five, and six lines were used as late as 1600.