720 BD screencaps are here: http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/
Before I forget, a HUGE HUGE thank you goes out to Reichu's Eva Commentary page that discusses Eva's script jargon. Having that resource seriously saved my ass with this translation. (For whatever reason, googling on animation or storyboard shorthand didn't return much results.)
Tokyo-3 has been FUCKED. Positively chilling stuff, in my opinion. If NME eventually gives us a Third Impact ending like EoE (instead of this Near Third Impact business) I wonder if the apocalyptic damage will manage to be several times worse than what EoE implied... This movie, whenever the hell it comes out, is going to be INSANE.
I proof-read my translation, but if I still managed to screw something up, any corrections would be greatly appreciated.

Enjoy!
Edit: Corrections provided by Reichu and symbv.

_________________________________
☆
○Tel Op "Preview"
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h50m29s162.png]

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Cut-1
A lake made out of the Geofront. On the lake's surface, a spear is struck through Eva-01. Track Up. The dark pilliars are lined up behind.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h50m41s136.png]

"Rei and Shinji still trapped"
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Cut-2
Close UP on the eyes glowing faintly.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h50m47s215.png]

"within the frozen Eva-01."
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Cut-3
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h54m22s48.png]

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Cut-4
Abandoned and changed city. A group of buildings, similar to a mass tombstone. Track Back. Telephone poles, among other things, have collapsed.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h54m30s140.png]

"The fortified city abandoned."
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Cut-5
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h54m37s218.png]

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Cut-6
V-22 Osprey descending on an aircraft carrier within an anchorage.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-11h54m51s79.png]

"Nerv officials are imprisoned."
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Cut-7: (Mentioned in the Storyboards, but NOT in this final script!)
Closeup of a register of names posted on a door with a chain across the front.
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Cut-8
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h00m45s37.png]

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Cut-9
Mark.06 descending to Dogma. Track Back.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h00m58s174.png]

"Dropped into Dogma"
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Cut-10
UP on that. (Face's part.)
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h01m20s46.png]

"is Eva-06."
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Cut-11
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h01m28s219.png]

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Cut-12
Eva-08 (floating). There's something like an angel's halo below it. Track Back.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h01m36s52.png]

"The quickening Eva-08 and"
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Cut-13
UP on that. The light of the 3 eyes come on. Track Back. (Slightly.)
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h01m44s131.png]

"its pilot."
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Cut-14
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h01m54s225.png]

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Cut-15
Kaworu's face, side profile, UP.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h02m01s43.png]

"Finally assembled"
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Cut-16
Ahead of Kaworu are four people's shadows extended. Track Back.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h02m11s123.png]

"are the children plotted by fate."
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Cut-17
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h02m21s249.png]

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Cut-18
UP on Misato's head. She's pulling on something to elevate it.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h06m47s70.png]

"In the end,"
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Cut-19
Ready with a gun, Kaji frantically shouting.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h06m55s162.png]

"for those people"
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Cut-20
Gendo and Fuyutsuki.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h07m02s236.png]

"who wish to live,"
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Cut-21
Rei waist-shot.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h07m20s163.png]

"where will"
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Cut-22
UP on Mari. As she removes her glasses, she glares in this direction.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h08m22s11.png]

"the story lead to?"
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Cut-22B
○Tel Op
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h08m30s79.png]

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Cut-23
Asuka's restriction support pillar has become empty. Light has also vanished.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h09m04s179.png]

"Next Time"
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Cut-24
Asuka raises her face.
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h09m23s100.png]

"Evangelion New Movie Edition"
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☆
○Tel Op "Next Time Evangelion New Movie Edition: Q"
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=vlcsnap-2011-05-24-12h09m37s253.png]

"Q. Next time as well I'll give you service, service~"
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☆
○(C) Khara-
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Edits and fixes as needed:
Edit # 1: Removed an unnecessary url-link.
Edit # 2: Fixed Eva-08 information; SmartOCR rendered the sentence structure a little funny and I brainfarted on how to read the line. Also fixed a line about Eva-01. Thanks, Reicchan and symbv!

Edit # 3: Changed "Advance Notice" to "Preview". Derp on my part with super-literal wording.
__________________
TV 2.0, Q/3.0's Next Time Preview:
For archival purposes, I'm following the thread's namesake and posting the images (from the Internets) alongside the text:
[URL=http://s74.photobucket.com/albums/i267/sailorstardust/Rebuild%20stuff/Q%20Next%20Time%20Preview/?action=view¤t=1314367254523.jpg]

Narrator Misato: "Shinji Ikari awakens. Beside him stands a mysterious boy. And then, a new world waits for him. Next Time: Evangelion New Movie Edition: Q. Next time as well I'll give you service, service~"