Born in Norwich, Wiggins was apprenticed at 14 to his uncle, Joseph Potts, the Sunderland ship owner. He was master of a ship by the age of 21 and owned cargo vessels himself. He married a Sunderland girl. For a time after 1868, he worked as a board of trade examiner in navigation and seamanship in Sunderland.
It was now that his lifelong obsession took over - the possibility of establishing a trade route between Western Europe and Asiatic Russia by way of the Arctic ocean and the great Siberian river systems. Wiggins fitted out a small ship at Dundee (103 tons) and set sail on 3 June 1874. The celebrated Henry Morton Stanley had been anxious to accompany him but was unable to do so. Wiggins reached the mouth of the River Ob after a difficult journey. The route had been used by Russian traders as far back as the sixteenth century, but Wiggins' voyage focused new attention on the commercial possibilities.
Wiggins sailed to Northern Russia in 1875, and again the following year, when he reached the River Yenisei and left his ship there for the winter. He set out to retrieve it in 1877, accompanied by Henry Seebohm, the ornithologist, but the ship was in poor condition and was soon wrecked. Wiggins' most brilliant feat was to reach up the great river as far as Yeniseisk in 1887-88, far above the supposed limit of navigation for such a large ship. Other expeditions failed, but in 1893, Wiggins delivered a cargo of rails for the Trans-Siberian railway at the mouth of the Yenisei.
On 8 August 1894 Wiggins commanded a small flotilla of ships that left the Tyne and set sail for the Kara Sea. It included the two tugs built at Low Walker, Pervy and Vtoroi (meaning 'First' and 'second'). They were joined in Norway by a wooden steamer, the Stjernen which was to return the crews to Britain. Wiggins delivered the tugs to the Russian authorities at Lokovoi Protok some 500 miles up the Yenisei. Unfortunately the Stjernen was wrecked off the Yamal peninsula, but with the help of Samoyed tribesmen and a Russian trader, Wiggins and his men were able to cross 600 miles of Tundra to Pustozersk on the Pechora river, and on 15 December they reached Archangel.
In 1905, during the Russo-Japanese war, famine was rife in Siberia and the Russian government asked Wiggins to organise and lead a large relief expedition by sea. He did his best, but was too ill to accompany the flotilla. He was buried in Bishopwearmouth.