The council awarded the £8.4m contract to Lakehouse in April 2017 and work started on site in November 2017.
The former owners of Lakehouse sold the construction division for £500,000 last summer to new owners Mapps Group Ltd.
The Enquirer reported in December that subcontractors were quitting Lakehouse sites over unpaid bills.
Hackbridge Primary school was one of the sites affected.
Sutton Council said: “The appointment of Willmott Dixon follows the council issuing a termination notice on the previous contractor due to performance and project delays.”
The ex-owners of Lakehouse – now rebranded as Sureserve – revealed last month that they have set aside £2.5m for possible claims on projects carried out by the new owners.
Sureserve confirmed it was facing “potential contingent liabilities arising from the disposal” including “potential claims under parent company guarantees and bonds for projects.”