The conn would have fallen either to Mr Worf, or to Mr Data.
...Lt. Data is 2nd officer... he would take over after Picard and Riker
Lt. Cmdr Data, then Lt. Cmdr Laforge, and sadly enough of the the Common Bridge Crew, Troy is next in line. She has an honorary Rank of Lt. Cmdr.
Dr. Beverly Crusher also holds the rank of Commander, and is authorised to command the bridge, but generally does not do so in favour of remaining in sickbay and allowing the designated command crew to take charge
Data is shown to take over as acting captain in Gambit after Picard and Riker are captured. Intriguingly, Worf is made acting first officer, suggesting he's next in the ship's hierarchy. The only times LaForge commands the ship are in the first season, when he was only a lieutenant junior grade.
Data is the Captain of the USS Enterprise-E in 2409. He has been since Picard resigned commision and became ambassador to Vulcan.
Before you point out Data was destroyed in Star Trek: Nemesis, read this from the sixth paragraph: http://www.startrekonline.com/node/183/long
^ That's B-4 with Data's memories. By 2409 the E has been retired and the F is in service, under Captain Va'kel Shon. And STO isn't canon anyway. - Mitchz95 17:42, March 17, 2012 (UTC)
Lieutenant Commander Data is Second Officer and Operations Officer and, as such, is third in command.
Troi and Crusher both happen to be staff officers and not line officers, but their rank of Lieutenant Commander is not honorary. Troi gets a taste of command at one point and subsequently decides to take the field examinations in order to become a line officer, and does. Crusher never does, but Picard places her in command at one point anyway. ProfessorTrek 02:08, July 1, 2011 (UTC)
(Someone fill in which episodes are applicable)
Beverly Crusher had the night watch every once in a while, so she knew what she was doing in TNG: Descent.
In the hierarchy, it goes Picard, who is captain, Riker, who is first officer, Data, who is second officer, and then Worf who is third officer, although Troi, Crusher and La Forge out rank two of then (by the last season any way) they aren't bridge officers, as such they don't have the authority to take over unless there are no higher ranking bridge officers, this is the same on modern navel ships, and it is even stated by Paris in that Voyager episode when they all get kidnapped.
Data is third in command of the Enterprise and, as such, he is in command when Picard and Riker are absent. If you watch TNG, then he is introduced to visitors as "third in command". A common misconception is that Worf's position as tactical officer or bridge officer gives him a higher position in the chain of command than LaForge. In "Lonely Among Us", Worf tells Geordi that the Captain wants his "junior officers to learn, learn, learn" and Geordi replies, "not just his junior officers." In Peak Performance, the episode specifically states that Geordi steps aside to let Worf act as first officer. Furthermore, Geordi is promoted to LT then to LT CMDR years ahead of Worf, which gives Geordi seniority. In addition, the position of Chief Engineer is by definition a senior officer. Just look how many times Scotty took command when Kirk and Spock were off of the ship. In fact, some naval proceedures place the Chief Engineer's authority over that of the XO. In addition, Troi and Crusher are staff officers, meaning that under normal circumstances they do not have operational authority, nor are they in the chain of command. Crusher specifically stated this in TNG: Encounter at Farpoint. Of course there are exceptions. For example, if the Captain orders someone to take command, then they have that authority regardless of their rank. In TNG: "The Arsenal of Freedom" the captain decided to give command to LaForge over Chief Engineer Logan, even though Logan was next in the chain of command. (Which is why LaForge has to keep telling Logan to back off.) Since the Captain is the final authority on how to manage his crewmembers, then he could have even assigned command to Wesley if he wanted to. Admiral William Thomas Riker 01:58, March 17, 2012 (UTC)
Actually Crusher holds the rank of Commander, not Lieutenant Commander, while it is not explained in canon, the novels infer that she held Bridge Officer qualifications before joining the crew.-Cpthunt 04:13, March 17, 2012 (UTC)