The bullets for the NGSW program were 125-135 grains depending on source. With muzzle velocity also depending on source, but always over 3000 fps. So I'd guess it would be about 130 grains.
The .277 Fury looks to be about the same size as legacy short action rounds, I assume the AICS mags they are showing are the standard size. I remember from earlier articles on Sig's NGSW submission that they are using steel case heads and running higher pressures to meet the Army's velocity requirements. Just playing around with .270-.308 in quickload, you'd have to run ~83 ksi with 130gr bullets, and ~90 ksi with 135gr bullets to hit their advertised velocities in a 16" barrel. It would take a topped out .270 WSM to hit those numbers in a 16" barrel at normal-ish pressures.
I'm guessing the case has a little more capacity than a .308, but I'll be interested to learn more about the cartridge.
Edit: The Sig rep in this video confirms that the .277 Fury is a .308 based round, and states that the performance spec of 3,000 + fps out of a 16" barrel is with a 140gr bullet. That's STOUT, above .270 WSM performance!
More info from "Tactical Life":
"Currently, SIG is moving forward with civilian versions of its 6.8mm cartridge developed for the [Next Generation Squad Weapons] competition, the 277 SIG Fury. This cartridge utilizes a hybrid case technology made from brass and other materials. The combination is rated to handle 80,000 psi of chamber pressure.
This added pressure will drive a 135-grain bullet from SIG’s Hybrid Match cartridge 3,000 fps from a 16-inch barrel, and produce 2,694 ft.-lbs. of energy. The 135-grain bullet has a respectable .488 G1 BC.
SIG’s 277 Fury Hybrid Hunting load features a 140-grain bullet with a .508 G1 BC, launching from a 16-inch barrel at 2,950 fps. That one produces 2,706 ft.-lbs. of muzzle energy, too.
One of the benefits of creating hybrid case technology for SIG is that the case is 20-percent lighter in weight for a loaded cartridge than a standard brass-cased cartridge."
https://www.tactical-life.com/gear/ammo/277-sig-fury-hybrid-case-design/