Anna
I can't see it being with any of Nelson's descendants.
After his death the majority of his possessions went either to Emma (his uniform coat etc.) or William, who of course had all the medals and decorations.
Everything that Emma had was sold.
Everything that William had was passed to his daughter Charlotte and then her son Viscount Bridport. (Except we know now that William gave away one very important item - the Star of the Order of the Bath.)
Bridport then sold all (to my knowledge) the Nelson artefacts in July 1895.
I can't recall any mention of this robe in any of the inventories, sales etc.
Another one to log as an unresolved mystery I guess.
I assume this is the robe for the Order of the Bath that Nelson is wearing in Gilray's famous cartoon.
MB