The Presbyterian Church of England was formed in 1876 by a merger between the United Presbyterian Church and other Presbyterian congregations in England. So there is no inconsistency between Arthur Macarthur being described as United Presbyterian in 1871 and simply Presbyterian in 1881, and no need to speculate about how many Presbyterian churches there were in Blackburn.
I have in my tree the Reverend John Black, born in Newcastle-upon-Tyne in 1831/1832 to parents from Lanarkshire, who attended, but did not graduate from, the University of Glasgow, and then took a theological course at the London Presbyterian College. He was minister of North Bridge Street Presbyterian Church, Monkwearmouth, Sunderland from 1859 until 1878. Following the merger of the United Presbyterian and other churches to form the Presbyterian Church of England, he was appointed the first General Secretary of the Presbyterian Church of England in 1878, and he served in this capacity until his death in 1888.
So it is at least a possibility that your Arthur Macarthur followed a similar path, for example attending the University of Glasgow but not graduating. It would be worth asking the University of Glasgow Archives if they have any record of him.
https://www.gla.ac.uk/myglasgow/archivespecialcollections/I see from a newspaper report of the laying of the foundation stone of a school in Blackburn that the Rev Arthur MacArthur is the only participating minister without letters after his name, which does indeed imply that he had never graduated from a university.
His death was reported by the
Yorkshire Post and Leeds Intelligencer on 14 July 1920:
The Rev. Arthur Macarthur, who for many years was minister at the Presbyterian church of Etal, Northumberland, has died in Newcastle the age of 77. The
Alnwick Mercury reported on 19 October 1912 that:
On Sunday, the Rev Arthur Macarthur preached farewell after having prosecuted his labours for over twelve years. Advancing years had had something to do with it. Interesting to see that another Rev Arthur Macarthur was the was the last Moderator of the English Presbyterian Church before its union with the Congregational Church in 1971.
Digression re the London Presbyterian College - see
https://dissacad.english.qmul.ac.uk/sample1.php?detail=achist&histid=66&acadid=180. I don't think it is likely that Arthur MacArthur attended this college, first because it was partly established to provide English candidates for the Presbyterian church in England, and second, why would he have gone all the way, and incurred the expense, for theological training in London when there were theological colleges in Scotland?
