Britain was by far the leading shipbuilder during this period. By the late 19th century, the two great centres of British shipbuilding, and by extension of the world, were the Clyde and the North East Coast. The North East ports had a higher combined total, but shipbuilding on the Clyde was far more concentrated.
To illustrate, in 1883 Clyde launched 417,881 tons, compared to 577,746 from the North East Coast. The latter figure was however combined from several locations, principally 216,573 and 212,313 tons from the Tyne and the Wear, respectively. Several other, more minor, locations were also counted in the North East total, including 81,795 tons from the Tees.
Both regions rose to prominence during the Second Industrial Revolution by leveraging their comparative advantage of proximity to iron and coal mines for building steamers. Hence, about 3/4 of the ships built at Clyde in 1883 were steamers. Thereafter production grew at a breakneck pace in subsequent decades, reaching 770,000 tons from the Clyde and 660,000 from the Tyne and Wear by 1914. By this point over 90% of British shipbuilding were steamers.
Out at the Clyde and the North East Coast were so high that Britain thoroughly dominated shipbuilding throughout the period. German production was ramping out rapidly near the end of the long 19th century, centred in the ports of Hamburg, Bremen and Stettin, but they were still dwarfed by Britain's. I don't have total production figures available, but for merchant shipping, British shipyards was producing 2,000,000 tons per year by 1914. In contrast, and for reference, German production reached 400,000 tons in 1914. French merchant vessel production only amounted to some 140,000 tons, and the United States, 135,000 tons.
Obviously, none of them could hold a candle to either Clyde nor the Tyne and Wear.
Jones, Evan R. "British Ship-Building During 1883." Commercial Relations of the United States with Foreign Countries, No. 37. US Government Printing Office, 1884.
Clapham, John Harold. The Economic Development of France and Germany, 1815-1914. Cambridge University Press, 1936.
Lerner, Williams. Historical statistics of the United States, colonial times to 1970. Part 2. Bureau of the Census, 1975.