Use light-gas guns.
Light-gas guns are basically a very powerful version of air guns. The idea is that you have a piston that compresses a light gas (usually hydrogen or helium) and then a rupture disk that breaks when the light gas reaches a specific (and very high) pressure. The light gas then blows through the rupture disk and accelerates the projectile down the barrel.
We typically use gunpowder in our world to drive the piston that compresses the light gas, but our gun powder isn't magic and doesn't kill us. In a world where gunpowder is problematic, just use something else to compress the light gas instead. A spring, steam, or even hand cranks with really large gear reduction ratios. Other than for fire rate, it doesn't really matter how long you take to compress the light gas. The rupture disk won't rupture until it reaches the right pressure. Once technology advances, electric motors could be used to drive the piston (possibly via hydraulics.)
To put into perspective what can be done with these, the United States Air Force and NASA use them for accelerating projectiles up to somewhere around low Earth orbital velocity for simulating impacts in space (e.g. from meteoroids, missile interceptor kill vehicles, etc.) For example, Range G at Arnold Engineering Development Complex can accelerate projectiles around 8 inches (20 cm) in diameter up to around 7 km/s (15,650 mph.)
Barrels of AEDC Range G. Source Wikipedia from USAF, public domain
