Let me unpack your question there, Uri.
When did μπ stop being pronounced [mp] and started being pronounced [mb], with voice assimilation? Early. It does not occur in Southern Italian Greek (them saying [panta] instead of [panda] for “forever” really sticks out), but it does everywhere else in Greek, and it’s a change that could have happened before /b/ without a preceding nasal went to /β/ > /v (which was in place by 1st century AD). I don’t doubt that we’re talking 1st millennium AD for /mp/ > [mb].
When did μπ start being used to transliterate /b/ in foreign languages, just as ντ started being used to transliterate /d/? Late. The giveaway is initial μπ, which violates Ancient Greek phonotactics. I wrote on the various transliterations of Bagdad in Byzantine sources, at Nastratios in Pagdatia. They uniformly use either Beta (anachronistically), or Pi for the initial /b/. Looking at the Lexikon der Byzantinischen Gräzität, initial μπ for loanwords with /b/ seem to become routine only from the 13th century on, though there look to be sporadic earlier instances.