Sanskrit: a Language Without Script
Sanskrit language was never a language of masses. It was an artificial language created to safeguard the interests of Vedics. They banned others not only from learning it, but even from listening the language. If Sanskrit words were listened by a Shudra, he was punished by pouring boiling lead in his ears.
If we look at the history of Sanskrit, one thing we clearly notice that Sanskritists avoided to use it in written form, and for centuries, it was just an oral language. The method of transforming the literature was to memorize, recite and handover orally to next generation. So they did not require any script as they did not require the language in written form.
But non-Vedic Indians used their languages in written forms also. This tradition of non-vedics goes back to Indus valley Civilization.
In Jain mythology, Rishabh, the fist ford maker of Jainism taught his daughter Bambhi to write. As Rikhabh was a pre-Vedic personality, this story indicates that pre-vedic Indians knew to write.
The oldest rock inscriptions in India are in Desi (i.e. Prakrit) languages, and not in Sanskrit language.
Vedics learned to write from non-vedics. Obviously, they used the scripts which were used for indigenous languages by indigenous people of India. That is why Sanskritists use different scripts in different regions of India for Sanskrit language. They also used different scripts for Sankrit language in different period.
The oldest written line in Sanskrit is a rock inscription at Junagarh in Gujarat, which belongs to 3rd century of Common Era. It is written in Bambhi (Brahmi) script. This script originally belongs to Prakrit languages, because all the oldest inscriptions in Prakrit languages are in this script, and all the oldest inscriptions in this script are in Prakrit languages.
A remarkable thing is that the oldest known Sanskrit inscription I have mentioned above was written at least five centuries after the oldest inscription in Prakrit inscriptions.
Later, Kharoshti and Gupt scripts were used for writing in Sanskrit language. Both the scripts were evolved from Bramhi script. Sharada, a script evolved from Gupt script was also used for Sanskrit for some period. In Eastern parts of India, Bengali and Udia scripts were used for Sanskrit, and in Southern India the Sanskritist used Kannada, Tamil, Telugu and Malayalam scripts.
The use of Nagari script for Sanskrit language started after 11th Century C.E. But this script was being used for Desi languages long before 11th century. The oldest inscription of Marathi language is in Nagari script, which was written in 981 C.E.
In 18th century, European scholars started to use Nagari script for Sanskirt language in large scale. Simultaneously, they started to use Roman script also. Today also, scholars use Roman script for Sanskrit.
All this proves that Sanskrit language does not have it's own script. If any languages does not have it's own script, nothing is wrong in it. But Sanskrit lovers are doing one wrong thing by saying that Nagari script originally belongs to Sanskrit language. In fact, Sanskrit has been always written in scripts borrowed from other languages.