Yikes.
There was a power loss in the system, I cannot recall offhand the common architecture for these systems, however eventually on the downwind the crew advise that they have no LLZ displays (ILS later noted). On a bad day on the east coast that is going to make for a busy cockpit. ATC maintained calm throughout as they developed understanding of the issue.
Having identified within the first 10 minutes or less that they were getting a repeat of the first event, then the crew response is calm, probably too calm. At that point they need to be taking immediate action to preserve their fuel status and increase their options as more information is gained on the tactical situation. Flying around at 2000' is not healthy. The crew did propose Stewart, which never seemed to be resolved. At around this time, KPIT was suggested by ATC, and that is 295nm away, but was VFR. At that point, the aircraft was approaching minimum fuel to divert to KPIT, and would have been down in the 3T remaining or so if proceeding there. An immediate emergency declaration on finding a repeat of the same instrument indications would have set up for a divert to a VFR airport. The aircraft instead travelled on various headings, East and otherwise for a period, and then a GPS based approach was flown to KEWR 04R. At the time the LPV/LNAV/VNAV approach was flown, the aircraft probably broke out at or above minima. It would not have had an alternate at that point, it was already on the divert however, so the OPSPEC would come into play to determine if the flight remained in a fuel emergency or not. I would suspect that it did.
Stewart used to have PRA capability, so if you have an ADI, heading etc, you can shoot an approach, assuming that you have some level of familiarity to the approach type. The LNAV/VNAV approach, would have placed the aircraft almost every time more reliably in the position of the final approach compared to an ILS that is not CAT II/III.
An interesting CRM exercise in a LOFT, and an event that many airlines can learn from.
Being too quiet about your issues may be great on the read backs, but being in a critical corner, and burning options out the exhaust from politeness may not help the reliability of the safe outcome.
The crew did a good job in trying circumstances, ATC did an excellent job. Once the wheels fall off the wagon, then what the crew do is up to their command judgement. Once you are in an emergency condition, all bets are off, and the only requirement is to minimise the risk to the outcome. At the time of the failure of the primary navigation system, and assuming that it could not be fault found and corrected with the QRH (which is highly likely, the ELEC, and INST sections of the B777 NNCL are brief, some knowledge of the bus architecture goes some way, as does talking to company engineers, or via them to the manufacturer), thereafter the viability of a SBAS/GPS LPV/LNAV/VNAV approach becomes seriously important. This was an emergency at that time, and the non ILS approach has the capability to achieve same or better performance than the ILS.
Perhaps as a professional group, we do not fly enough approaches using the alternative to ILS. I routinely would fly an ILS approach using the LNAV/VNAV tracking in the B744 and the B777. It does an excellent job. Even without any of those aids, flying a track and FPA (its a B777... easy to do) from a known distance or fix will keep the plane on the approach within normal tolerances. Yes, we used to demonstrate that routinely as well. Do I recommend flying that as a primary method? Nope, but having some faith in the systems capabilities at least gives some options to be followed when the wheels fall off the wagon. Having formated large aircraft on other small and large aircraft, there are always some options that may be considered, depending on background and how big a hole the operation is in. A similar event occurred and ended badly in another part of the world, where the crew of a jet did an approach and missed at minima, and lost the alternate at that time. an hour later, they parked their jet in the water, and survived. They were conducting approaches in an emergency condition, with some excellent capability, but never transitioned into an emergency authority frame of mind. After going around at PUBLISHED minima on multiple occasions, they parked the plane in the water. Many years before, at the same airport, my plane had the same situation. We declared an emergency, descended out over the water to visual conditions, and tracked on the available precision navaid (not so precise... doppler and airborne radar) back to the airport (1,050' below the minima for the approach). We had to climb back into the cloud to cross the cliff, and then sighted the runway threshold, and dumped the sorry bird on the ground. We had already briefed for a ditching as a potential outcome. Once you are in an emergency, your options change. Today, we have better tools, and the options may be easier.
My views are not a criticism of the flight crew, they did a good job. They could have had less stress if they had declared the emergency, and dealt with it as a real emergency at that time. From a training point, understanding the capability of the systems and what backup you may have at least gives a quick set of options to consider, which makes the decision making process less traumatic.
ATC-Cockpit communications were really pretty good in the circumstances.
Overall, good outcome, potential good training value to other crews for SA and ADM/NDM heuristics. I would fly with the crew. I would also buy the ATCO a beer.