Just to touch on definitions first, nuclear and chemical waste refer to different types of hazards.
Nuclear waste is the usually the remnants of spent nuclear fuel, or byproducts in the processing of the the fuel before it is used. Nuclear waste is hazardous because is radioactive (emits particles that are dangerous as they damage DNA).
Chemical waste is usually from other industrial processes, and is dangerous because the chemical itself is harmful when ingested, inhaled, etc.
Nuclear waste has varying degrees of severity- high level (very dangerous) waste includes spent nuclear fuel like caesium 137 and plutonium, to low level waste like clothing that has been contaminated with radioactive material. Depending what element it is, the half life (how long it takes for the radioactivity to reduce by half) could be decades to thousands of years.
Nuclear waste can be flammable, which is why many waste facilities invest heavily in remote fire detection and extinguishing. However, it depends on what type of waste- just as some materials are not flammable, not all waste is either. Intermediate level waste is often formed in to a 'sludge' before being placed into containers which would likely be less susceptible to fire, for example.