No, you cannot use GoDaddy's privacy services for a .co.uk. If you would like privacy you will need to "opt out" You can find more information on opting out at Nominet's website: http://www.nominet.org.uk/other/whois/faq/
Yes, you can Opt-out with all .CO.UK domains, and it is free. You should be able to gdo it from godaddy's website... however... if they have blocked it you will need to get a security code from them through the post: