I'm thinking of moving a .co.uk site from its UK server to a US one. Will this affect its search engine rankings in the UK?
You server's geographic location do affect your local search ranking but if you consider all the variables you wouldn't even notice any difference.
It would effect your ranking between in Google.com and Google.co.uk due to server geo-location changed. - If your domain is .co.uk extension, your ranking in Google.co.uk may down a little while Google.com may be moved up a little - If your domain is TLD (com net biz org info), your ranking Google.co.uk may down like a rock while Google.com may be moved up as a high jump.
I don`t think that is true Local search ranking is about the language...for example, if your site is in french, you will have good rankings in France...it doesn`t matter where the site is hosted
Your SERPs should not be highly affected by switching server. Only impact that i see is that there could be a slowdown in the crawl rate as there could be a downtime or a different IP serving website which never happened before. After few days or couple of weeks even this should settle down. Local searches do not depend upon the location from where the pages are being served rather depends upon your language selection or if you have adjusted your preferences thorough webmaster tools.
Totally no. But it affects only if you where using a co.uk and change it to .com but if it is just the server hosting no problem, no worries.