The SARS-CoV-2 variant B.1.1.7 reportedly exhibits substantially higher transmission than the ancestral strain and may generate a major surge of cases before vaccines become widely available. As B.1.1.7 can be sensitively detected using the Thermo Fisher TaqPath S-gene RT-PCR test, contact tracing and isolation programs appear well-suited to slowing the spread of the new variant, which is still rare in most of the dozens of countries in which it has been identified. However, key determinants of outcomes such as data-sharing, trace success, and isolation compliance vary widely between regions, which may discourage public health agencies from explicitly redirecting existing contact tracers to contain B.1.1.7. Here we apply a branching-process model to estimate the effectiveness of implementing a B.1.1.7-focused testing, contact tracing, and isolation strategy with realistic levels of performance. Our model indicates that bidirectional contact tracing can substantially slow the spread of B.1.1.7 even in regions where a large fraction of the population refuses to cooperate with contact tracers or to abide by quarantine and isolation requests.