Do we know how long it took for cuneiform to develop from counting cows and barley, to drafting official documents and contracts, to creating literature?

The best way to get a good answer is probably to ask the man on anything cuneiform, Dr. Irving Finkel.

...and then to register astronomical observations! The birth of science, no less.

And all because every year like clockwork, the Eufrates and Tigris blanketed an area of hundreds of square kilometers with a fresh coat of silt (from the Taurus mountains in modern-day Turkey) that was perfect as a rudimentary but cheap, easy and quick writing medium, pushing the point of a stick into a pancake of soft clay, then leaving it to dry and harden in the sun.

Another way of putting the question:

What was the process of ancient Sumerian experimenting and realizing the potential of this new tool they had on their hands?

"Maybe I can use writing to do such-and-such thing..." and proceeded to do the first ever written snail-mail message, or the first medical or mathematical instructions/textbook, the first poem or essay, etc.