Hi,
my website is aimed at the European and UK market. I am considering hosting it with a US webhost, mainly for reasons of cost and quality of service. Will doing so harm my search engine rankings for people searching from the UK and Europe?

Thanks

No, it would only have a marginal influence. A lot more important are standard SEO practices, like title tags, headers, website copy, link building. If you're worried about it, you could do a search on some of the keywords you're interested in and see what hosting your competitors use. (you can use http://www.whoishostingthis.com ) or better yet check what is really important, and that is how many backlinks they have. (I recommend http://tools.seobook.com/firefox/seo-for-firefox.html for this)

P.S. Since you're from the UK http://www.hosttell.com/one are currently having a promotion in the UK. They are offering one year of a free hosting and a domain (You have to pay the £9.50 setup fee and that's it). They're a nice choice if you don't need more than 3GB of disk space.