You have seen that I asked a question about Proto-Indo-Uralic and whether or not it could be reconstructed using Proto-Indo-European, Proto-Uralic, and internal reconstruction in those branches. One person said kind of, and the other one said no. (I'm reading the paper that one of the people sent me about Schleicher's fable being reconstructed in PIU.) However, if such a language existed (and I believe so and more importantly there is evidence that it did, though the proof is not settled nearly as much as the Indo-European or Uralic languages), when was it spoken?
Short answer: about to 6001 BC near the northern beaches on Caspian Sea.
As @jknappen states, there is probably no evidence that Indo-Uralic really existed. But if this paper is believable, it states that:
Based on the current archaeological and genetic data, it is likely that the Neolithic Pontic-Caspian steppes represented the Proto-Uralic community to the west (Mariupol) and the Proto-Indo-European community to the east (Samara-Orlovska), already separated during the 6th millennium BC; before, during and after which period they influenced each other with successive population movements. We will assume in this paper an ancient genetic relationship—that is, that Early Proto-Indo-European is in fact Proto-Indo-Uralic—which is supported by the initial formation and continued similar genetic admixture in the Eneolithic steppe. By the time of the Suvorovo-Novodanilovka expansion at the end of the 5th millennium, though, they were already two different, unintelligible languages.
Based on this, we can assume two things:
- Proto-Indo-Uralic as one language is probably spoken on 7th millenium BC.
- Since the 6001 BC is the latest 7th millenium BC date, the last time Proto-Indo-Uralic was spoken as the same language was around 6001 BC.
Indo-Uralic is probably spoken near the northern beaches on Caspian Sea. If the Wikipedia is believable, it states that:
The Dutch linguist Frederik Kortlandt supports a model of Indo-Uralic in which the original Indo-Uralic speakers lived north of the Caspian Sea, and the Proto-Indo-European speakers began as a group that branched off westward from there to come into geographic proximity with the Northwest Caucasian languages, absorbing a Northwest Caucasian lexical blending before moving farther westward to a region north of the Black Sea where their language settled into canonical Proto-Indo-European (2002:1). Allan Bomhard suggests a similar schema in Indo-European and the Nostratic Hypothesis (1996). Alternatively, the common protolanguage may have been located north of the Black Sea, with Proto-Uralic moving northwards with the climatic improvement of post-glacial times.
There is currently no evidence for an Indo-Uralic clade in linguistics, see this question and its answers for a discussion of the current state of Indo-Uralic.
There is probably a last common ancestor for Uralic and Indogermanic languages, but it probably was also the ancestor of some other language groups as well.