Boris Johnson said on Tuesday that he could not “see any reason” to deviate from the road map, which would see further lockdown restrictions lifted on May 17 in England.
For people who test positive, a “comprehensive” self-isolation support service is available which includes payments of up to £500 for residents on low incomes.
The DHSC said the operation is the “largest surge testing operation to date”, aimed at suppressing any possible new cases of the variant, with positive results from PCR testing to be sent for genomic sequencing at specialist labs.
It said all those who have tested positive for the variant, with the first case in the area being found in early March, are isolating or have completed their isolation, and their contacts have been traced and asked to isolate.
Director of public health for Lambeth, Ruth Hutt, said: “It is vital that all Lambeth residents take an asymptomatic PCR test so that we can identify all cases in order to stop the spread of this variant of concern.
“Alongside taking a PCR test, we are asking everyone who lives and works in the borough to continue following the guidance in place to stop the spread of Covid-19.
“That includes limiting your contacts with other people, washing your hands regularly and thoroughly, keeping your distance and covering your face.
“If you test positive for Covid-19 via a PCR test or rapid lateral flow test it is crucial that you self-isolate so you don’t pass on the virus, and support is available to help you do that.”