"I completed the current objective, but my squad can�t be ordered to go to the next one and there�s no obvious way to proceed."
This happens at a specific point in the mission. You�re supposed to head back to base at this point, but the game instead incorrectly tells you that you need to talk to your squad sergeant to give orders. Just leave the area on foot, heading out the way you came. Or if you want to save time and think you can do it safely, jump over a few obstacles and climb down from the roof, then continue back to base that way. After a short distance, you�ll be teleported back automatically, at which point the squad will resume behaving properly.
"Can I access this from a saved game made prior to the DLC pack�s release?"
Yes. At almost any point in the game, you can go to this new quest and it will work fine even if you�re using an old save. The only point of no return is the final room of the final quest, at which point you obviously can�t go back. If you start a new game instead, the quest will become available shortly after you leave Vault 101.