AnnOMaly:
Let's say the neighbors knew who the new boss was very quickly, by mid-July. They would have to meet up and confer with everyone concerned (Ezra 4:7-10), compose and translate a letter,
They had already sent complaints to the previous king. Why would further 'conference' be required? The scribe could just write substantially what they'd previously written to 'Ahasuerus'.
send the letter to (presumably) the Persian capital Susa, allow time for the Persians to dig around the archives for historical information on the Judean kingdom, allow time for the Persians to compose a reply, have it sent back to the local Samaritans' or neighboring peoples' officials and have it read out to them before rushing over to Jerusalem to implement the order.
Or Badiya already didn't like the Jews and lied about the research. Or the research had been done during Cambyses' reign. It's not really clear why the writer of Ezra (or 1 Esdras) would have in their possession a letter sent by their enemies to Persia's king and the Persian king's response to their enemies anyway, so it's possible that some or all of the content of either letter was simply made up. There's many instances in the Bible where conversations - even private conversations among enemies - are simply made up where the writer could not possibly have been party to the conversation.
Bardiya was dead by the end of September. Soon after Jerusalem got the reply, he'd be history. His order to stop building could hardly be enforced long-term, so why did the Jews wait to finish repairing the city and its walls until (the next) Artaxerxes' 20th year - 78 years - which made Nehemiah depressed - especially if you believe Darius gave the go-ahead to rebuild the city?
Maybe they were just lazy. Apparently they only did any rebuilding in ernest when there was an order to do so. I'm not sure that Nehemiah's depression would be a particularly great motivator to Jerusalem's populace.