Both natural infection with SARS-CoV-2 and immunization with vaccines induce protective immunity. However, the extent to which such immune responses protect against emerging variants is of increasing importance. Such variants of concern (VOC) include isolates of lineage B.1.1.7, first identified in the UK, and B.1.351, first identified in South Africa. Our data confirm that VOC, particularly those with substitutions at residues 484 and 417, escape neutralization by antibodies directed to the ACE2-binding Class 1 and the adjacent Class 2 epitopes but are susceptible to neutralization by the generally less potent antibodies directed to Class 3 and 4 epitopes on the flanks of the receptor-binding domain. To address the potential threat posed by VOC, we sampled a SARS-CoV-2 uninfected UK cohort recently vaccinated with BNT162b2 (Pfizer-BioNTech, two doses delivered 18-28 days apart), alongside a cohort sampled in the early convalescent stages after natural infection in the first wave of the pandemic in Spring 2020. We tested antibody and T cell responses against a reference isolate of the original circulating lineage, B, and the impact of sequence variation in the B.1.1.7 and B.1.351 VOC. Neutralization of the VOC compared to B isolate was reduced, and this was most evident for the B.1.351 isolate. This reduction in antibody neutralization was less marked in post-boost vaccine-induced responses compared to naturally induced immune responses and could be largely explained by the potency of the homotypic antibody response. After a single vaccination, which induced only modestly neutralizing homotypic antibody titres, neutralization against the VOC was completely abrogated in the majority of vaccinees. Importantly, high magnitude T cell responses were generated after two vaccine doses, with the majority of the T cell response directed against epitopes that are conserved between the prototype isolate B and the VOC. These data indicate that VOC may evade protective neutralizing responses induced by prior infection, and to a lesser extent by immunization, particularly after a single vaccine dose, but the impact of the VOC on T cell responses appears less marked. The results emphasize the need to generate high potency immune responses through vaccination in order to provide protection against these and other emergent variants.