To give some background, I was taught classical Sanskrit at school and understand the language of Rig Veda passably. I opened Avesta transcribed in my mother tongue ( Gujarati - since the majority surviving Zoroastrian community used to speak that language ) on archive.org out of curiosity and found it quite legible. I am not sure of the composition time or geographical area of its composition. But I was struck by how similar it sounded.
I am not sure it is possible to answer this question in any definite sense, but first: the similarity between the Sanskrit of the Rig Veda and Avestan is a key point in dating the oldest part of the Avesta, the Gathas, to around the same time (c:a 1300 BC) as the Rig Veda. So it's not actually surprising that you found it legible! As always, for further reading on Zoroastrian traditions, the work of Mary Boyce is recommended.
I can provide you with one more thing that suits your timeframe for Iran - livius.org has transcriptions of the Royal inscriptions in Achaemenid Persian. (e.g. the Behistun inscription of Darius the Great ), so you can see for yourself (at least some cognates are fairly obvious - "khshayathiya" (transcribed xšâyathiya) and kshatriya; vazra- and vajra-, etc). It's worth noting that Iranian languages were also spoken by the Scythians, Alans, and other nomadic groups roaming the steppes north of the Caucasus and Caspian sea. However, these languages aren't well recorded in the timeframe you specify, which makes intelligibility much harder to assess.
I'm afraid I can't help you with regard to the Indian subcontinent though.
It's pretty clear that Sanskrit had a huge impact on the local dialects in many regions of the Indian subcontinent. If you were to travel south to the Deccan plateau, or east towards Patliputra you would find many people (mainly brahmins) speaking Sanskrit, or a slightly modified version of it.
I am unsure about Persia and Western Asia.