Localization For Two Different Regions/Domains with the Same Language

I'd use WPML plugin, it allows tranlations of posts, pages, categories, menus, and plugins that have language files. It will let your end use login from the same admin area, and then add translations to the existing content.

The tricky bit will be setting it up to work on domain detection, normally it works on a subdomain, or add's a variable to the end of the URL, like ?lang=uk

But that said it should be possible though with a bit of tinkering to setup something based on the URL. And the documentation is pretty comprehensive. Good luck!

It's also possible to make it differently. For our shops we pretend there's a new domain, but in fact there isn't. This also works for SEO because we do make sure the language tags and such are correct. We replace only a few parts such as a few javascripts and conversion pixels and then we are done. See for example on Keepershandschoenen-shop.be. This site is in fact a clone so we don't need wpml. We don't even need a wordpress, magento or whatever CMS installation...

So then you have the same backend, with .com acting as .co.uk also.