Is a Hajj performed by my uncle considered incomplete, and what should he do now, after he took an umbrella and a prayer rug that he found discarded on the ground during Hajj, and returned with them to his country, thinking there was no harm in that?

What your uncle found in Mecca is considered lost property (Luqata), and the rulings of lost property apply to it. There is no difference between lost property found in the Haram and elsewhere, nor between lost property found by a pilgrim and by others. It is permissible for the finder to take it, and he is obligated to publicize it. However, if it is insignificant and of no value, and people's souls do not pursue such things, then there is no harm in him benefiting from it without publicizing it. It appears that the mentioned items are of this type.