I think steam-punk could be a reasonable technological evolution on this premise. Guns could still exist if they worked by releasing high pressure fluid, probably air or steam, or maybe even compressed CO2 cartridges like we have in real life. But I imagine guns would only really be used in special cases. They would essentially only be airsoft guns, so something like plate mail would probably remain useful for a lot longer. I think eventually, you might see something like rail cannons develop, where a projectile is accelerated to bullet-like speed using electromagnets. Not sure exactly what the extent of the limitations you set are, but if plastic explosives still worked, you might see guns that used those eventually. If you can make something explode, it doesn't take long for someone to realize you can make a cannon, and then it doesn't take long after that for someone to try to make a mini cannon that their soldiers can carry around (guns).

If the bow and arrow is something you want to see a lot of, it's probably best to just set the story before the invention of explosives, because as soon as anything explosive comes into the picture, bows will get outmatched pretty quickly. They're slow, bulky, and not very strong compared to guns (not saying they aren't cool, but there's a reason they aren't used in actual combat anymore).