The number of new daily SARS-CoV-2 infections experienced an abrupt increase during the last quarter of 2020 in almost every European country. The phenomenological explanation offered was a new mutation of the virus, first identified in the UK. We use publicly available data in combination with a time-delayed controlled SIR model, which captures the effects of preventive measures and concomitant social response on the spreading of the virus. The model, which has a unique transmission rate, enables us to reproduce the waves of infection occurred in the UK. This suggests that the new SARS-CoV-2 UK variant is as transmissible as previous strains. Our findings reveal that the sudden surge in cases was in fact related to the relaxation of preventive measures and social awareness. We also simulate the combined effects of restrictions and vaccination campaigns in 2021, demonstrating that lockdown policies are not fully effective to flatten the curve; fully effective mitigation can only be achieved via a vigorous vaccination campaign. As a matter of fact, incorporating recent data about vaccine efficacy, our simulations advocate that the UK might have overcome the worse of the CoVid-19 pandemic, provided that the vaccination campaign maintains a rate of approximately 140k jabs per day.