Review in your shops' staging environment that the Nosto tagging refers to the new site domain and hostname meaning that your site maps Nosto the new correct urls.

As a precaution, just before you are doing the changes and changing settings on your shop, temporarily disable Nosto’s features which might refer to the old shop name

Go to account settings and change the frontpage URL or add the new domain as an accepted host.

Request a manual, full re-index which will update URLs. Once update is finished, the person who launched the re-indexation will receive a notification email

Review with debug-tool that everything works normally and enable Nosto again!