Further notes and observations after reading the last several posts:
ThreeDee, why did you pick Tautog for this scenario? On 07 December, Tautog was in port at Pearl undergoing refit. In the OTL she did not get underway until the day after Christmas. Unless you chose Tautog for a specific reason, it may make more sense to have one of the boats that was already at sea to intercept the retreating Japanese. Sister boat Trout and the big old Argonaut were off Midway, with Tambor and Triton patrolling north of Wake. One of these boats would probably be more likely to intercept Kido Butai than the Tautog, which probably wouldn't have been able to get underway for several days at best.
Radio reception for low frequency (LF) was possible at periscope depth using a loop antenna. No part of the boat needed to be above the water, indeed the loop antenna itself was usually 10-15 feet under the water at PD. Transmission or HF comms was a bit trickier, requiring a whip style antenna to be above the water. There were a few different types of these antennas with one type attached to the periscope. The height of the antenna above the water determined the range of transmission, with very long range requiring the boat to be surfaced. Assuming your boat is relatively near Hawaii, a contact report could be sent with the boat broached (i.e. only partially surfaced).
As to the chances of sinking a carrier, refer to my post above. There are a lot of variables to consider and the idea is at least possible. They would have to be in a favorable position close to the enemy track and ahead of them, and they would have to get through the destroyer screen without being detected (probability low to medium). At this point approved tactics dictated that they dive to at least 100 feet (they are most likely already there) and track the target on sonar alone. Optimum position for firing would be (obviously) ahead of the target and between 1000-2000 yards off the target track (projected course). Once satisfied with the firing solution they would set the running depth of the torpedoes to approximately 30 feet so that they will run under the keel (the optimum spot for the Mk 6 magnetic influence exploder to detonate the warhead) and fire off a full spread of six torpedoes. They would then go deep (250 feet), rig for silent running, and wait to hear the detonation of the weapons, all the while expecting a depth charge attack from the escorts. But given the conditions that existed within the submarine force at that time, I believe the likelihood of this type of attack being successful to be quite low. During the war, this type of approach was used 31 times with no known successes. Sonar bearings were just not accurate enough and range was nothing but an educated guess. Both of these elements are EXTREMELY important to accurately aiming torpedoes. Sonar was very useful in the initial detection of the target and detection ranges out to 15,000 yards were possible under good conditions. But the very fine and accurate tracking necessary to aim torpedoes was just beyond the capability of the relatively primitive systems of the day.
If Joe Willingham made the unlikely decision to toss out the rule book and make a daylight periscope approach, the likelihood of success goes up considerably, from less than 5% to about 10-30%. Uncertainty and hesitation were quite prevalent and caution was the mantra of operations in the first few months of the war. It would have taken a truly gutsy skipper to go against everything he had been taught was right and to toss out all the experience he had (even as flawed as it later turned out to be) at this stage. Indeed most of the reports of the first submarine actions amply demonstrate this unfortunate mindset. It took time to build up the experience and confidence necessary to take greater chances.
The Mk 14 was actually a very well made, precision-crafted piece of machinery. But the damn thing was full of bugs and faults due to a stunted R&D phase and the overconfident intransigence of the managing engineers at the Bureau of Ordnance. It ran too deep due to the problems I outlined above and this prevented the Mk 6 magnetic exploder from firing when it should. When this fault was discovered, the crews simply set it to run shallower, but then discovered that the MK 6 exploder was so sensitive it set off the warhead prematurely, before it even reached the target! So then they just deactivated the magnetic feature and set the weapon to impact the side of the target using the regular contact exploder. But due to a fault with the firing pin, the damn thing wouldn't go off and it bounced off the side of the ship! The BuOrd managers steadfastly refused to admit that anything was mechanically wrong with the torpedo, blaming all the troubles on the boat crews. Only after field tests and trouble shooting in theater revealed and corrected the problems, and only after being ordered by higher authority to pull their collective heads out of their arses did BuOrd finally get onboard and fix the issues at the factory. When combined with unrealistic pre-war training and poorly thought out tactics the torpedo problems reduced the submarine force to frustrating near impotency.
These issues are why I believe that the chances of getting in a successful attack on one of the Kido Butai carriers at this time to be very low.
As for how this scenario would effect other battles, I just couldn't say for sure. Lots of butterflies there.
The Tautog is there because of butterflies from a TL running in the background that puts it exercising NW of Pearl on the 7th. CAPT Charles Lockwood is onboard the Tautog observing the exercises (this is important because he is getting firsthand experience of the failures of their tactics and torpedoes).
So the scenario seems to be improbable on multiple levels, and I don't think I will be using it in my TL. My goal with this event was twofold; to through a series of high profile events make the higher ups in the Navy realize and take action about the faulty Mk 14 torpedo and submarine tactics, and to make the Japanese fleet seem vulnerable to submarine attack.
I thought of an alternative scenario that seems from what you've said more probable.
The attack begins, and the Tautog receives word almost immediately. It makes all haste East (instead of NE as in the original post). It estimates the fleet's location using the attacking aircraft's bearings and encounters the fleet due East just as they're leaving northward. Willingham prioritizes alerting Pearl of the fleet's location, bearing, and speed over using the "proper" tactics, and stays at periscope depth, firing simultaneously. From periscope depth, they are able to effectively aim and get off a good shot. The message is sent and received, and the Tautog begins an emergency dive to escape their ASW measures. There are four premature detonations and two hits, with both failing to detonate. The Tautog launches its aft torpedoes from the "proper" depth and miss horribly. They stalk the fleet for as long as possible, but they are vastly slower, and quickly fall out of range. The US fleet tries to intercept them with this information, but are unable to.
This still accomplishes my goals and does not cancel the invasion of Wake (which is the next important event).
Does this seem more probable?