You have calculated things ghastly incorrectly. The technological development of these people would never reach the medieval era. Metal has been the primary fuel driving the technological evolution. Without metal, these people would be stuck in something between the dark and neolithic ages.
You are right that battering rams can be built with wood and stone. But how are you going to cut the wood so precisely? With obsidian knives and axes, right? Yes you can, but obsidian blades get blunt very quickly and you have to chip them off again and again very soon until they get too thin to be useful anymore. Plus, obsidian is a rare substance, found only in volcanic regions. Read about the technological level of the Aztecs and Mayans. That should give you some insight into how advanced (or not) a civilization can get without metal. And then again, Aztecs and Mayans did have pure gold and silver to use, and in the past, had evolved Asian people no farther back in time than 12000 years ago or so.
Anyway, that was a tangential discussion about how advanced your people can get. Now back to the topic of warfare without metal.
Talk about wooden clubs. The royals and officers would have their clubs spiked with obsidian and rock. That is going to be your primary war weapon at close range.
Another weapon would be short leather whip, with shards of sharp rock and obsidian glued to it.
Atlatls would be very common for launching javelins. While atlatls usually tend to decrease the accuracy when used for maximum range, a whole legion of atlatl throwers would do devastating damage as hundreds of atlatls are launched from distances of about 100 - 150 meters from both sides.
Also, there might be legions of people adept in throwing knives. While the range of throwing knives is far restricted than those of javelins, they would still be used for sudden hit and run tactics and guerilla attacks.
Yet another weapon which would get a lot of importance is the blowpipe and dart. Accurate, but having short range, it would be another weapon of choice for guerilla attacks and diversion tactics.
The bow and arrow would (obviously) be the most powerful and accurate weapon. It would obviously be impossible to develop the powerful modern composite bows, but classical composite bows such as Mongolian and Turkish would be possible. These would require a lot of time and effort to build, but would be highly prized weapons of choice in a war. Considering that metal is not available, arrowheads would be either made of hardwood, horn, thick plant thorns or carefully chipped rock.
Poison would be the secret to winning wars. Fast acting, debilitating (in a battle, fast action is more important than lethality) poison recipies (such as those incorporating aconite and strychnine) would be kept secret and highly prized.
Poison would be applied to all types of range weapons, javelin and arrow heads and on the ends of darts. Wars might be decided not on the basis of numbers, but on the basis of which party has faster acting poisons.
I don't think your people would be able to make clean, factory grade gunpowder, because of the requirement of hydraulic press, which can practically only be made with metal. Your folk could concoct a crude mixture of black powder (potash nitrate and coal), but the type required for use in canons would be impossible.
It might be possible to make some type of a very crude (and extremely short lived) canon or matchlock guns, but that too, is extremely unlikely as making smooth holes in long, thick logs is extremely difficult and error-prone effort in a metal-less world. Those people would not have the pipes available easily to start experimenting with them in the first place, to invent canon.
Black powder would be quite useful in bringing down fortifications though. And in guerilla attacks where it would be used as distraction device and to blow up enemies in ambush attacks.