Finch has been chief executive of Centre for Cities since it was set up in 2005. Previously, he was a policy adviser at HM Treasury for 11 years.
He will move to Fishburn Hedges in August, replacing Lucy Burns, who is taking a career break.
He said: 'This is a fascinating time to move into public affairs, with enormous policy challenges facing the new Government and difficult decisions facing private and non-profit players.'
Centre for Cities is a research and policy institute that helps cities improve their economic performance.