The present disclosure relates to methods for processing liquefied waste polymers containing diolefins and naphtha. Hydrotreatment A of the diolefins produces a diolefin depleted LWP feed which is separated by distillation B to give rise to one or more distillates containing at least a naphtha fraction, an optional middle fraction and a distillation bottom. Hydroprocessing C of the naphtha fraction gives rise to hydrogenated naphtha fraction which is suitable as a feed for a steam cracker D. The distillation bottom and the middle fraction can mixed with crude oil and used in oil refinerery since they are predominantly free from diolefins.