Do I have a hope of driving organic UK web traffic to a US based website?

Hello there - long time lurker, first time poster.

I work for a global software company as a regional Marketing Manager. The company is HQ-ed in Germany so has a strong market presence there - it also has a strong presence the US and France. As a consquence, our English website is in the United States, and web visitors can change their language (not location) via the drop down at the top - we currently have English, France, Germany, Span and Japan. UK web visitors are directed to the US...

Do I have a hope of driving organic UK web traffic to a US based website?