New data from PHE suggests 10,400 deaths have been saved by the vaccine up to the end of March, using a 31 day lag between vaccination and a mortality endpoint.

They've calculated this by looking at actual mortality and vaccine coverage at any point.
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https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/977249/PHE_COVID-19_vaccine_impact_on_mortality_March.pdf

It then ratios up actual deaths, to get a "without vaccine" expected deaths figure, looking at age bands separately. It makes no allowance for "second order" effects such as the vaccine reducing transmission and thus bringing down prevalence. It assumes 81% efficacy.

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The paper is able to use actual vaccine figures, an efficacy assumption for the single dose strategy, and also implicitly the method allows for the lockdown bringing down prevalence. But otherwise it's the same concept as used in this predictive paper.
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Both papers show the first effects starting around Jan 10th, and building as the pace of the roll-out accelerated. Without trying to predict lockdown effects the @Aneas_Journal paper assumed constant prevalence and expressed the falls as %s of a without vaccine situation.

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The main difference is that without evidence of first dose efficacy @Doctimcook and I assumed 100%, although included a sensitivity at a lower rate of 90%. Similarly, take-up was assumed to be 100%, the sensitivity of 95% for >80s and 90% below appears reasonable to date.
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REACT yesterday also showed divergence between the fall in infections and deaths, showing how the vaccine is layered on top of the lockdown effect in bringing deaths down even faster. The effect is less for hospitalisations, again consistent with the @Aneas_Journal paper.
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It's great, if not surprising, to see actual evidence of the vaccine working, but it only works in the arm not in a fridge, so addressing continuing hesitancy is the best way to ensure protection for all communities.
https://opensafely.org/research/2021/covid-vaccine-coverage/#weekly-report
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