Without putting too much thought in and no doubt breaching the criteria.
Lancaster to Edinburgh. Runs generally South to North but from West to East coast. (The Lune is tidal under the railway bridge at Lancaster which is the last bridging point)
Glasgow to Plymouth? Glasgow probably doesn’t meet the criteria due to the Erskine bridge further down the Clyde but this would link West, East, West again at Bristol and South Coasts.
Dumfries to Newcastle. Again, Dumfries is inland and Newcastle not quite coastal but you certainly see the sea on the west coast on the journey then go what most would deem across the entire country.
In times past there was a Sunderland to Whitehaven.