Projects paid for by the public and deemed to be benefitting the whole UK include an extension of the London Underground, a bridge in Liverpool, the Lower Thames Crossing in London, the Thameslink and Crossrail networks in London, various roads (most notably including the A14 from Warwickshire to Suffolk, but none in Scotland, as road-building in Scotland is a devolved responsibility) and of course the HS2 rail line, speeding passengers from as far north as Birmingham to (surprise!) London.
@ willy wombat. If we're talking public funding of transport related infrastructure - I'm glad I'm not an English taxpayer living outside the south east:
Wings Over Scotland | The pooling and the sharing