It depends on the variant. A preprint suggests that the vaccine may be as effective against B117, the variant that has arisen in the UK, as it was against the earlier forms of the coronavirus.
But against the variant circulating in South Africa, B1351, a preprint suggests it may be much less effective at preventing mild to moderate disease than it was against previous forms of the virus. It’s believed this is because of a mutation called E484K, which changes the virus’s exterior so that existing antibodies to SARS-CoV-2 are less able to bind to it.
Sarah Gilbert, lead researcher on the Oxford Vaccine Development Programme, believes that the Oxford/AstraZeneca vaccine will still protect against severe disease when facing these variants. However, this has yet to be proven.
Variants circulating in New York (B1526) and Brazil (P1) also have this E484K mutation. We don’t yet have data on how well the vaccine works against these variants, but it’s plausible that it may struggle against them too.
In the meantime, the vaccine’s developers have confirmed that they can tweak it to handle these mutations. They estimate an updated booster could be ready by autumn.