Good way to change from entirely different template in Joomla 2.5

I am planning to move the entire Joomla 2.5 website from a template that is not made by you to OT Winter template. Both templates have major differences in module types, sizes and locations.

What is the best general strategy to gradually move the content to the new template:
1)Without affecting the current live site; 2)Without modifying the directory structure so related Google search result links give no 404 errors; 3)If possible, without setting up a completely separate development.