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To be able to show the same reviews on the carousel of both store, you would need to export the reviews from the UK store and import them onto the US one.
We do have a feature available only in the Awesome plan, called Cross-Shop Syndication (you can learn about it here), that serves to sync and share reviews across stores, but it doesn't support the carousel. So if you want to show the reviews in the carousel the only option would be to export from the UK store into the US one.
On the UK store, you can export the reviews in the Import/Export settings here:
Then before you import the reviews into the US store, you will need to edit the export file to change the product identifiers. These are the product ids and product handles and are used to correctly match the reviews with products. The export file will contain the product identifiers of the UK store, so you need to change them to those of the US store.
To learn how to get the product identifiers, refer to our guide here:
For Shopify: Get your Shopify product ID and product handle
For Woocommerce: Product Identification Values to Import Reviews
When you edit the file, I recommend using Google Sheets instead of Excel, since it tends to break the file's CSV format.
Once the file is ready to import you will be able to upload it into your US store, here:
If you decide to import/export the reviews, don't use the cross-shop syndication feature at the same time, because you will end up showing duplicated reviews on both stores on all the other widgets.
We have both a UK and US shopify website. In terms of the judge.me carousel reviews - Once we publish those on our UK site is it possible to sync/transfer them to our US site too so they appear there too?