According to an article by Guy Norris in Aviation Week, the gear box issue is holding up deliveries of the Trent 1000 powered aircraft as Boeing needs replacement gear boxes from Hamilton Sundstrand. There are numerous 787s on the Boeing flightline waiting to be delivered. Of the 9 airplanes that I have listed as in final preps for delivery at Everett, 5 are equipped with the Trent 1000 engine. According to Guy Norris' report, 17 Trents in total are affected by the gear box issue of which nine are on undelivered 787s that are earmarked for ANA. Boeing and Rolls Royce are waiting for replacement gear boxes from Hamilton Sundstrand. Of the other 4 787s at Everett/Ft. Worth that are waiting to be delivered, one is in paint (ZA262), one for JAL (ZA182) just emerge from the paint hangar and will robably start pre-flight work soon, the one for Qatar Airways is still finishing up interior IFE work (ZA461) and another 787 that just appeared at the Boeing fuel dock, looks to be United Airlines first 787 (ZA288) and still has to go through painting before moving on the pre-flight. All these are equipped with the GEnx engines.
So it appears the bottleneck may be with the gear box issue with the Trent 1000 engines as each of the 5 Trent 1000 aircraft are all at different point in the pre-flight timeline. Each aircraft has to wait it turn while the aircraft that is further along must clear each pre-flight milestone before the B-1 flight.
In the meantime, the Trent 1000 gear box issue is likely a minor problem that affects a very small portion of the 787s in service but it's an issue that has attracted significant attention because it is the 787.
Lastly, LAN says it is expected to take delivery of its first 787 (ZA536) in late August or early September.