Alright, people! You've caught my attention!
Now, here's my thoughts on the matter, using the old instructions manual (Windows 95 version, not DOS) for references.
Let's dial this back to 1996, when the game first came out, and for the sake of argument, let's consider weapons that only existed around that time.
And let's also be realistic: so, having a trooper carrying an M134 Minigun with an ammo bagpack and feed belt might work in theory, but this guy would literally have to be Arnold Schwazzenegger (as he is the only person I know of who has actually lifted a Minigun...which is one reason why they gave it to him in Terminator 2-because he was the only person who could actually lift the heavy SOB); so giving it as standard-issue equipment to every infantryman is completely impractical. They'd be too slow, they'd be big fat targets, and I don't see a fire team of these guys fitting into APCs.
Alot of assessments here are very good so far!
So, if you're talking infantry only, on GDI, you'd see your infantry use NATO standard weaponry based on their home country. Expect to see riflemen carrying Calico M-950 Semiautomatic Rifles (U.S., according to imagery), SA-80 L85A1 Automatic Rifles (United Kingdom), G3s (Germany), AK-47s (Russia), and the like. Grenadiers of course would employ hand grenades, like the M61 or M67, but in modern war, hand grenades can only go so far, so infantry would likely employ the M203 40mm Grenade Launcher underbarrel attachment, M-79 breech-loaded 40mm grenade launcher, or if Russian, the GP-25. Pistols can be an assortment of M9s and M1911s.
Light machine guns might be too exclusive to elite units at the start of the war, given GDI's budgetary and political constrains, but you might see infantry packing the American M294 SAW or M-60, the British SA-80 L86A1 Light Machine Gun, and Russian PKM or RP-46/RPK.
I'd like to think that some technology might exclusively be used only by the Elite Forces Unit (i.e., Rangers, Commandos, Special Forces) at the start of the war. So, even M-16s for Americans might be exclusive only to commandos. Other tech like M-4s, M-82 Barretts, certain optics, thermite, and even some types of body armor might be kept in reserve for the better soldiers, not out of some kind of spite, but because GDI can't afford to give every soldier of its so many variant nationalities a standard weapon systems (save the Calico...useless 9mm piece of crap, no penetration power whatsoever...). I say this because the instructions manual claims the Commando was armed with a "Raptor .50 cali assault rifle" and an unnamed "Sniper rifle." The only Raptor in C&C was the 100-round assault rifle in Renegade. The only .50 cal assault rifle I can think of is the Beowulf, but that didn't exist in 1996. So, you can be liberal and give the Commando any weapon that would fit his/her mission parameters. As for sniper rifles, I'd go with the suppressed PSG-1, Remington M700, M40A1.
As for shotguns, I'm going to go with the Italian SPAS-12 or the M590A1 Mossberg; throw the Bayonet on there for added close-quarters measure.
For antitank detail, you're going to see troopers packing the M-72 LAW and M136 AT-4. Russian soldiers of course might use the RPG-7. Later in the war, you'll see soldiers packing Redeyes or FIM-92A Stingers for antiaircraft assignments.
On Nod side, you'll see troopers armed with a myriad of weaponry based on their home country. In particular, IRA personnel (according to the book, Nod was significantly connected to the IRA, and given the circumstances of 1996 when the Provisional Irish Republican Army broke the truce with the British government early in the year, it leads to good recruiting opportunities for Nod), you're going to see AR-18 Widowmakers in use by IRA personnel. Otherwise, Nod will use anything they can get their hands on, mainly old Soviet technology.
For the more exotic weapons, like flamethrowers, you might go with the American M9A1 or Soviet LPO-50.