Couple of huge mistakes in your post. Sanskrit never was language of common people, it died thousands of years ago. No one speak sanskrit as mother tongue in India, only 10.000 speak as 2nd language for religious reasons.
Sanskrit was spoken not just in India but across SE Asia. In fact, languages such as modern Malay and Bahasa Indonesia have evolved from a Sanskrit base. As time went on, Sanskrit evolved on different lines and depending on geographical proximity to other Sanskrit speakers and intermediate influences. So Assamese, Bengali and Oriya sound similar with multitude of common words even though the Oriya scritpt is different from Assamese / Bengali. (BTW, the modern Thai script originated from the Oriya script). Similarly, Singhalese evolved differently from other North Indian Sanskrit influenced languages as its intermediate influences were Pali and it used the Brahmi script.