Make sure there are no project errors with either the template or variants.
If you're uploading variants from a Google Sheets spreadsheet, make sure you've shared it with the Ads Creative Studio service account. The email address is unique for your Ads Creative Studio account, but it will end with @partnercontent.gserviceaccount.com.
Make sure the spreadsheet has Variant Id and Variant Name column headers.
Make sure the spreadsheet's swappable element column headers have the same names as the swappable elements in the project you're uploading to.
If you've checked all of these issues and still can't upload, it may be a problem with Ads Creative Studio. Try waiting a few minutes, then upload again. If it's still not working, contact support.