I understand that the variation selected is not being displayed in cart or checkout, is this correct?
I am unable to replicate this issue on my test site, as the variation selected is being displayed on cart and checkout, as seen in the screenshot below:
Cart:
Checkout:
I’d suggest running a conflict test to rule out any theme/plugin conflict. The best way to determine this is to keep your theme on Storefront, and temporarily disable all plugins except for WooCommerce.
Also, if you have any custom code (such as Additional CSS) on your site, please can you temporarily disable this as well.
Check if the issue is present at this point, if not it means there was a conflict, and you can reactivate the plugins one by one (gradually) until the cause of the issue is found.
You can find a more detailed explanation of how to do a conflict test here.
You can clone your site to a staging environment and perform tests without modifying your live site or impacting customers. This is possible to do it using the free WP Staging plugin.
If the conflict test makes no difference, please can you share a copy of your sites System Status Report which can be found by navigating to WooCommerce → Status. Select Get system report and then Copy for support.
Once you have this, please share the results in your response here.