The subordinate conjunction that originally appeared after the clause “saith the prophet”. The that would work better if the phraseology had been “for behold the prophet saith that the time cometh” since English readers expect “saith that” without an intervening subject. But in actuality, the clause “saith the prophet” acts parenthetically, with the result that without the parenthetical clause the sentence would read “for behold that the time cometh”. Such use of “for behold that S” (where S stands for a sentence) is quite unexpected in modern English, which is probably the reason why Joseph Smith removed the that in his editing for the 1837 edition.
A similar example, but with a different intervening parenthetical clause, underwent the same editing for the 1837 edition:
There are also examples in the original text where that directly follows an imperative behold:
The first example involves a conjectured it after nourished. For discussion, see Jacob 5:24.
We also have clear evidence that the subordinate conjunction that can follow a parenthetical “saith the prophet” or “saith the Lord”. In each case, the that complements the verb preceding the parenthetical clause:
These examples suggest that the original reading in 1 Nephi 22:15 (“for behold saith the prophet that the time cometh”) is indeed intended. The expression “for behold that S” is acceptable in the original text of the Book of Mormon.
Summary: Restore in 1 Nephi 22:15 the that in “for behold saith the prophet that the time cometh speedily”; the that does not complement the verb saith (which occurs in a parenthetical clause) but rather the preceding verb behold; Joseph Smith sometimes deleted the following that when behold occurred as an imperative (or as an interjection without a subject).