Hi Mark,
I'm not sure I really understand the question.
From my point of view www.domain.co.uk, domain.co.uk, and www.www.domain.co.uk are all host names representing root web sites, so when you say they installed it in a folder it makes no sense to me as no folder is reflected in any of those urls. Therefore I guess you mean they moved it to a different root site folder.
Moving the files from one root web site to another should not be a problem as long as file permissions are correct in the new location.
Skins and content are 2 different things, you do no need to do anything with the content, you just need to diagnose the skin problem. Maybe they didn't copy all the files correctly under /Data/Sites/1/skins
The way to trouble shoot it is check the files and also view the source of the rendered page, grab the url for the skin css, paste it into the browser url bar and find out if the css is delivered or an error is happening. Also review the mojo error log for clues. If there are not read permissions on the skin files in the new location or if the files are missing it could cause this.
See also notes about moving a mojoportal site from one server to another, the same issues apply if moving from one web site on a server to another.
Hope it helps,
Joe