Paraffin waxes are a promising material for heat storage with high energy density. Their low thermal conductivity, which limits the speed of charging and discharging in heat buffers, was previously shown to be improved by adding graphene nanofillers. In the present study, using molecular dynamics simulations, the segregation by molecular weight of polydisperse paraffin near graphene flakes is investigated. In liquid bidisperse paraffin composed of decane and triacontane, an aligned layer containing mainly triacontane was observed next to the graphene. Upon slow cooling, the wax crystallised into distinct layers parallel to the graphene sheet, with much stronger segregation by molecular weight than in the crystallised bidisperse wax without graphene. For polydisperse wax, the segregation effect was much less pronounced. The molten paraffin had a somewhat higher concentration of the longest chains in the first layers next to the graphene, but during crystallisation, the molecular weight segregation was only slightly increased. Measurements of crystallinity using an alternative version of the method developed by Yamamoto showed that the layers of wax were highly aligned parallel to the graphene, both in the solid state with all wax crystallised and in the liquid state with one layer of aligned wax above and below the graphene. Thermal conductivity was increased in planes parallel to the graphene flakes. The strong segregation of chain lengths in the bidisperse wax resulted in clear differences in thermal conductivity in the segregated regions. The less segregated polydisperse wax showed less variation in thermal conductivity.