I understand that you don't want them to have a different experience; however, you should want them to have a unique experience.
If you are just doing it for Google, I would read the link I included in my original post, which states, "...in some cases, content is deliberately duplicated across domains in an attempt
to manipulate search engine rankings or win more traffic. Deceptive practices like this can result in a poor user experience, when a visitor sees substantially the same content repeated within a set of search results." and "In the rare cases in which Google perceives that duplicate content may be shown with intent to manipulate our rankings and deceive our users, we'll also make appropriate adjustments in the indexing and ranking of the sites involved. As a result, the ranking of the site may suffer, or the site might be removed
entirely from the Google index, in which case it will no longer appear in search results."
Here is an example - SAP provides business software solutions globally. They have MANY versions of their website. Look at the subtle difference between content, images, etc. SAP-US and SAP-UK. They have the same type of pages on both sites, but the content and positioning is slightly different.
I understand that you are saying your visitors want the same end result whether that is a product or service from your company. You may just want to try having a landing page specific to the UK? You could design it like your homepage and then have it link to content throughout the rest of your site (the same stuff you are linking to for US based customers). You would then want to add domain.com/uk to all of the UK based business directories and such. Maybe create a Facebook page that is for that UK based domain?
Hope this helps. It is a good idea to target other areas... I am just saying that if you start duplicating your content, you could run into trouble with Google rather quickly.
Good luck.
Mike