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In English history, Conformists were those whose religious practices conformed with the requirements of the Act of Uniformity and so were in concert with the Established Church, the Church of England, as opposed to those of Nonconformists whose practices were not acceptable to the Church of England.
In 1662 ministers had to sign the Act of Uniformity to use the book of common prayer. Virtually the only church in the UK that is still truly "Conformist" in this sense is the Church of England (Continuing).