Ranking will differ according to search engine, enquirer's location and the exact search term. Supposedly strong backlinks from UK sites will strengthen your position in the .uk versions of the major search engines. Certainly you will get more useful direct traffic from UK links. I don't know about "regional" as opposed to "national" UK sites.
I experimented with two domains a couple of years back and found that the UK links worked better when searching specifically for UK supplier. It affirmed its UK standing. Those sites still do well today.
In my view common sense prevails! If a US site "vote" for a UK site then why would Google.co.uk value that as highly as a UK "vote" - all else being equal a UK based link far out weighs its US counterpart.