Trust chairman Sir Simon Jenkins, energy and climate change minister Ed Davey and HLF London chair Wesley Kerr unveiled the turbine, which is designed to help make Morden Hall energy self-sufficient.
The Dutch-built turbine will generate 59,000kWh of electricity annually, enough to power Morden Hall's visitor centre.
Following a redesign, the centre is claimed to be the most energy-efficient historic building in the country. It features solar panels, an air-source heat pump and a wood-burning stove.
The screw is also being funded by City Bridge Trust and Thames Water. The turbine was installed by Mackley Construction and Hallidays Hydropower.
Kerr said: "This is an innovative and exemplary approach to restoration, conservation and interpretation. The addition of the screw makes this a genuinely sustainable 'deep-green' heritage project, simultaneously bringing alive the industry of the past while embracing that of the future."
He added: "This whole scheme is an exemplar of what can be achieved in public parks."