Seeking McMullen or McMullan connection

Our ancestor Henry McMullen (or McMullan) was born in Armagh in1804, Parish unknown. Emigrated to Cumberland,England and lived in Wigton where he was married on 1824 in the local Church of England parish. He was a weaver. Later moved to Carlisle were the family was censed in 1841 and 1851. Their children in chronological order were Edward, Mary, Edward II, John, George, James, David, Anne, Samuel, Henry, James and Jane. According to the children naming practices of the time it is suspected his father was Edward and his mother was Mary ( John was the name of his wife's father and Anne his wife's mother).

Later they moved to Sunderland, Co Durham where he worked in a Marine Store and later for Customs and Excise and died in 1881. It is thoughthe had relatives in Cumberland and Sunderland.

I am aware of the many McMullens and McMullans in Armagh but never found Henry's birth records or the name of their original parish.

We will be very thankful for any suggestions.