This paper presents an advanced stochastic analytical framework to quantify the benefits of smart electric vehicles (EVs) and heat pumps (HPs) on the carbon emission and the integration cost of renewable energy sources (RES) in the future UK electricity system. The typical operating patterns of EVs/HPs as well as the potential flexibility to perform demand shifting and frequency response are sourced from recent UK trials. A comprehensive range of case studies across several future UK scenarios suggest that smart EVs/HPs could deliver measurable carbon reductions by enabling a more efficient operation of the electricity system, while at the same time making the integration of electrified transport and heating demand significantly less carbon intensive. The second set of case studies establish that smart EVs/HPs have significant potential to support cost-efficient RES integration by reducing: (a) RES balancing cost, (b) cost of required back-up generation capacity, and (c) cost of additional low-carbon capacity required to offset lower fuel efficiency and curtailed RES output while achieving the same emission target. Frequency response provision from EVs/HPs could significantly enhance both the carbon benefit and the RES integration benefit of smart EVs/HPs.