The original manuscript must have read “that they shall hide up their treasures” since Oliver Cowdery’s correction to that in the printer’s manuscript agrees with the that of the 1830 edition (here both 𝓟 and the 1830 edition are firsthand copies of 𝓞). The correction in 𝓟 appears to be immediate (there is no change in the level of ink flow for the supralinear that).
As a connector, either that or when can occur after the phrases “the time cometh” and “the day cometh” (here I include syntactic variants such as “the time shall come” and “the day shall come”), with 43 instances of that and 7 of when. Since either reading is possible here in Helaman 13:20, there would have been no grammatical motivation for Oliver Cowdery to edit the text in this case.
As noted under Helaman 13:16, both 𝓟 and the 1830 edition are firsthand copies of the original manuscript from at least Helaman 13:18 through the end of Mormon. In many instances, a corrected reading in 𝓟 will agree with the 1830 reading for this part of the text. In such cases, it will normally not be needed to discuss such variation since the text is firm and 𝓞 very likely read as corrected in 𝓟 and set by the 1830 typesetter. In some cases, the variation may be of some interest, in which case I will discuss it, but normally I will not. However, in volume 3 of the critical text, when I discuss the transmission of the earliest text, I will provide a list of all these cases.
Summary: Accept that as the connector after “the day shall come” in Helaman 13:20; the virtually immediate correction in 𝓟 agrees with the 1830 reading.