The youngest son of George Neilson, a calenderer, he was born in Glasgow on 24 September 1795. Educated at Glasgow High School and Glasgow University, he received a business training at offices in the city, and then joined his father in exporting cambric and cotton goods to America.
He also proposed some naval inventions. From 1822 to 1828 Neilson was in America on business. In his latter years Neilson suffered from heart disease, and he died at Kirkintilloch on 3 May 1861, and was interred in the burying-ground of Glasgow Cathedral.
In 1846 Neilson proposed improvements on the life-buoy to the Admiralty.
On 8 January 1848 he wrote to Lord John Russell, suggesting iron-plated ships, and 1855 he corresponded further on the subject with Lord Panmure and Admiral Earl Hardwicke. After the building of HMS Warrior and HMS Black Prince, Neilson summed up his views in Remarks on Iron-built Ships of War and Iron-plated Ships of War, 1861.
Shortly afterwards he published another pamphlet, on the defence of unfortified cities such as London.
