Did those feet in ancient time walk upon England's mountains green? (Origin of the story)

The short answer is William Blake. It is all in the reading of the poem and understanding Blake. He wished that instead of Israel, Jesus was born in England. Why were the Jews the chosen people and not the English? Because of the 'dark satanic mills'. Jesus will not come to England unless the social injustices are removed. After Blake a supposed legend about Jesus visiting England started. There is no earlier legend.

There are a range of interpretations of this poem. Speculation abounds, but one piece that is lacking in some of the discussions I have read is to recall that this is a preface to the poem Milton. Milton was non-sectarian in his theology, but clearly anti-Catholic supporting the Reformation in England. A movement intent on cleansing the hypocrisies in the Roman Church of that time.

As many great poets, Blake sees the past, present and future and envelopes these into one treatise. I believe that he, like Milton, longs for a pure expression of Christianity and posits that that purity will be present in the "New Jerusalem" which will be brought to England through the enterprise of the British people. The first two verses ask questions about whether the Lamb of God did visit England which are clearly intended to be answered negatively. Therefore if it is true, the Lamb of God did not visit England, then it is for us to bring the New Jerusalem to England through our own work.

With reference to the story of Jesus' visit to England with Joseph of Arimathea, OP asks:

• So how and when did the story get started?

  There is, so far as I know, one scholarly article on just this question which has recently had some scrutiny. The article is: A. W. Smith, "'And Did Those Feet...?': The 'Legend' of Christ's Visit to Britain", Folklore 100.1 (1989): 63-83. Smith works carefully, investigating a number of questions related to the central one, and so really needs to be read in full by anyone interested. The leading findings are these (citing Smith's own form of the questions, "it" refering to the legend of Jesus-in-England):

• What current form or forms does it take? How far back can it be traced?

  Smith documents a wide variety of forms of the story, with many details differing based on locale. They share, minimally, Joseph (usually as a metal merchant) coming to England, and Jesus coming with him. Beyond that, Smith finds the welter of changing detail to indicate a "weak and fugitive" tradition that is "far from" unvarying.

• How far is it the creation of identifiable literary and clerical authors, and how far rooted in oral folk tradition?

  Smith traces documentary evidence in sketchy form to the late 19th C, and much more from the 1920s, and attributed to a trio of energetic clerics who boosted the legend. One of these sources, H. A. Lewis, Christ in Cornwall? (3rd edition, 1948) is available online.

  Beyond that, Smith argued, it was a matter of hazy oral tradition and local lore, and that demonstrable only into the 1890s.

• Is the assumption that it "was known to William Blake and is alluded to in his lyric 'Jerusalem'" in fact correct?

  According to Smith, the short answer is "No" (see pp. 71-72). In his own words:

  [T]o return to our starting point, does Blake's lyric attest a knowledge of or a belief in a literal visit by Jesus to Britain? I would suggest it does not. I have searched Blake's poetry, prose and ephemera for any evidence that he knew this story and can find none.... (p. 72)

  Moreover, not only is there no written documentary evidence to suggest it, but Smith goes on to adduce some positive considerations explaining why it would be unlikely that Blake should be read this way.

• Is there any historical context or religious system of thought which could account for its origin and its spread?

  Smith thinks there is, and he locates it in the developing British Israel traditions of the growing Empire in the early and mid-nineteenth century. These, he suggests, picked up medieval hints along these lines (interestingly, associated with the apostle Paul), which mutated into the "legend" as we now see it (pp. 76-80).

I mentioned there was some "recent scrutiny" of Smith's article. It comes in a fascinating book: William John Lyons, Joseph of Arimathea: A Study in Reception History (Oxford: OUP, 2014). He devotes a chapter to "The 'Jerusalem' Joseph" which considers Smith's article at some length. Lyons writes cautiously of it, but (so far as I can see), is broadly approving. (Unlike his remarks on Dennis Price's fanciful The Missing Years of Jesus: The Extraordinary Evidence that Jesus Visited the British Isles [Hay House, 2010].)

But Lyons's work is ultimately pursuing a different agenda. Smith has assembled the data, sorted and sifted it, and considered the matter carefully -- whether he has placed the most compelling construction on the evidence he has uncovered may be up for debate, but it looks cogent to me.