The ‘Bible Christians’ were a Methodist denomination formed in 1815 and centered on Devon and Cornwall.

I think there’s a very strong possibility that miners from that area migrated to Cleveland to work in the mines and brought this specific brand of religion with them, as emigrants from Devon and Cornwall also took the religion to America and Canada.

The church name ceased to exist in 1907 when they merged to become the United Methodist Church which itself merged in 1932 to become the Methodist Church of Great Britain.