There doesn't seem to be any record of the two empires ever campaigning against each other. Of course records for this period are annoyingly scarce, and there wasn't a lot of time (perhaps 60 years or so) before Cyrus the Great swept in and conquered them both.
There are some records that seem to be saying this alliance was cemented by marrying Nebuchadnezzar to the Median king's daughter, but either way there's no indication of military hostilities against them. The places he was said to campaign after that were Syria, Arabia, Elam, Egypt, and assorted locales in the Levant. The enemies fought were Phoenicians, Egyptians, Jews, Arabians, Elamites, as well as periodic rebels in the various areas of the neo-Babylonian Empire he was putting together.
The only hint we have of any discord with the Medes during his reign was a mention of him hosting a "Median defector".
After his death, there were only 4 more kings before the Persian conquest, and only 2 of those served for more than a couple of years. Neriglissar only ruled for 4 years, and is only known to have campaigned in Anatolia. The final N-B king Nabonidus had a nice long 15 year reign with a rather abrupt ending. He was only known to have campaigned in Arabia starting in 553BC, which (probably not coincidentally) was the same year the final Median king started his reign and Cyrus started his rebellion against it. Nabonidus apparently liked it in Arabia, and stayed there for almost all of his reign, until just before it was his turn to get conquered. Cyrus invaded and added the Neo-Babylonian empire to his own in a quick campaign in 539.
