Josiah Bartlett was a Congregationalist.
He was identified as a Congregationalist by The Congregationalist Library. (Source: Ian Dorion, "Table of the Religious Affiliations of American Founders", 1997).
From: Robert G. Ferris (editor), Signers of the Declaration: Historic Places Commemorating the Signing of the Declaration of Independence, published by the United States Department of the Interior, National Park Service: Washington, D.C. (revised edition 1975), page 39-41:
Thanks to the voting order in the Continental Congress, Josiah Bartlett of New Hampshire was probably the first Delegate to vote for independence, the second to sign the Declaration (after President John Hancock), and the first to ballot for and pen his name to the Articles of Confederation. He also has the distinction of being one of several physician signers. His State service, more extensive than the National, included the governorship...Note that numerous sources and authoritative references have been consulted in order to ascertain the religious affiliation of the American Founding Fathers. Note that the excerpts and references mentioned on this page are not the only references used in order to identify this person's religious affiliation.
In 1794, the year before he died in Kingston at the age of 65, ill health forced his retirement from public life. His remains lie in the yard of the Universalist Church in Kingston.