Welcome to the Q&A Forum

Hi Joel, There are not much more specific things you can do.  The type of SEO you do that target US will also work in UK like creating content with keywords, on-page SEO, joining UK's social networks, sharing, backlinks and etc. One suggestion I can give is the language you use.  Although people speak english in UK but the way they spell may be different; for example, "optimize" and "optimise."  Same meaning but different spelling.  SO make sure your content includes the spelling people use in UK since that might be the way they search.

I dont think Demoting links will be a good idea!