Neglected Heirloom transforms once, into Ashmouth Blade, and stays that way. (Until something exceptional happens, like another effect or removal, like you said.)
The transformation rule on the Neglected Heirloom front face doesn't matter once it's transformed. The currently face-up side is all that matters, which is Ashmouth Blade. It doesn't have an ability that transforms it, so that's that. Neglected Heirloom's transform ability doesn't exist anymore.
There's several other cards that transform only one way, so this isn't entirely unusual, and for all of those the other face no longer matters. Ormendahl, Profane Prince doesn't have the mana-generating abilities of Westvale Abbey either, for example, and doesn't have anything available to transform it back into a building.
(Also, because of the "no flippy-floppy" rules change introduced with Shadows over Innistrad, that original ability on its front face can't transform it back and forth if the equipped creature happens to transform multiple times at once.)