I've seen many requests for such migration software. Since my current webshops are running on Prestashop 1.5.6 and I am moving them to NopCommerce as well as the lack of such migration software made me start up a new project for the smooth migration of PS 1.5.6+ to NC3.40
The migration software uses the PS API but since that API needs an https connection I have adjusted the PS php API software a teeny bit to easy the migration.
Off course the PS Theme will not be migrated, only your precious data.
My guess is the migration application will be ready by mid august.
Everybody who is interested, follow this thread.
Stefan