The player may move around while reloading, but using a Logos ability, switching to a different weapon, being affected by a stun or knockback or firing the weapon will interrupt the process. A weapon can be reloaded even if it is not completely empty. Attempting to fire a weapon or use a tool while it is empty will cause the character to automatically try and reload it. You will not be able to reload a weapon or tool if you don't have any spare ammunition of the correct type.
For most weapons and tools the time it takes to reload a weapon is static, but some types of weapons can have their reload time reduced with advanced Training. Two examples are Firearms Training for Pistols and Leech Gun Training for Leech Guns.