British Leyland had a plant in Belgium until the early 80s. They, and other manufacturers also used to ship cars "CKD" (complete knock-down) to several countries for assembly, including notably Australia and New Zealand. Land Rovers in particular were assembled in many countries.
British cars were also licensed to be built in several countries. For example, Austin Sevens were built as Dixis by BMW, and in the US as American Austin and later Bantam.
Rolls Royce had a plant in Springfield, Mass during the 1920s.
In the main, though, LBCs as you know them were mostly made in the UK. Britain had little in the way of raw materials, except coal, so iron ore, rubber, bauxite, etc. were imported, mostly carried in ships of the British Merchant Navy, which at that time was the world's largest.