Industrialization in New England began largely after the War of 1812, partly because of the war and partly for other, unrelated reasons.
The unrelated reason is that this just happened to be when Francis Cabot Lowell brought textile mills to America. In 1810, he toured factories in England. By 1812, he managed to set up his own textile mills in the United States. This timing had nothing to do with the war.
However, the war did help New England to industrialize. While the US was at war with Britain, there was no trade between the two countries. This meant New England manufacturers did not have to compete with English companies. This allowed them to grow rapidly during the period of the war and to continue growing after the war.
Thus, the war did help to allow New England industry to get on its feet.