Once Oxygen is activated, it disables the active theme on the site, which means that anything created or rendered by the theme will also be disabled. As such, it's not recommended to install Oxygen on live sites.
There are two methods that can be used to rebuild existing sites; the method you need to choose will depend on if the site is static or active.
Static Sites
If the site is static and doesn't have users logging in or purchasing products then you can create a fresh WordPress install, install Oxygen and rebuild the site from scratch. Once you're happy with it, you can follow these steps to make it live:
Take a full backup of the live site in case you need to revert your changes
Export the new site using All In One Migration and import it to the new site: https://en-gb.wordpress.org/plugins/all-in-one-wp-migration/
Once the site has imported, you will need to:
Save your permalinks twice via Settings > Permalinks
Resign your shortcodes via Oxygen > Settings > Security
Regenerate your CSS cache via Oxygen > Settings > CSS Cache
Live Sites
If the site is active and not static, you can do something like this:
Clone the site to a development/staging environment
Install Oxygen on the site clone
Create templates to display your content and rebuild your pages using Oxygen
Once you're happy with the design and that everything is functioning correctly, you need to merge the changes with your live site:
Put the live site into Maintenance mode
Take a full backup of the live site in case you need to revert your changes
Install Oxygen
Export Oxygen's settings from the development site and import them to the live site via Oxygen > Export & Import
Create all of your templates on the live site. Copy the shortcodes for the templates from the development site to the live site by clicking +Shortcodes below the Edit with Oxygen button
If you've added any pages or modified existing pages using Oxygen, you will also need to copy the shortcodes for those pages from the development site to the live site
Once everything is in place, you will need to:
Save your permalinks twice via Settings > Permalinks
Resign your shortcodes via Oxygen > Settings > Shortcodes
Regenerate your CSS cache via Oxygen > Settings > CSS Cache
If you have any caching plugins installed, you will also need to clear the caches.