The scheme involves thousands of contact tracers to track down people who will be told to self-isolate and determine who they have had contact with.
The aim of the system is to lift blanket lockdown restrictions and move towards more localised, targeted measures.
However, Exeter Labour MP and former government minister Ben Bradshaw, said that Baroness Dido Harding - who is heading up the wider test, track and trace programme - had told him on an MPs’ conference call that the system would not be "fully operational at local level till the end of June".
He added: "Not sure where that leaves [Boris] Johnson’s promise of a fully operational “world beating” system by Monday."
The government said on its website that the new system would "allow us to trace the spread of the virus and isolate new infections and play a vital role in giving us early warning if the virus is increasing again, locally or nationally".