In my opinion, the Dalai Lama is Avaloketishvara.
On that note, yes, he may consult Nechung, however I do believe that since the Dalai Lama has the attained clairvoyance to see many lifetimes into the future, he doesn't actually rely on Nechung.
It seems like it's all part of the fiasco. Why? If the Dalai Lama is not Avaloketishvara, he is still not a silly man. No one would say, "Don't rely on spirits", and have the state oracle take trance of a subdued spirit!
If he did believe in Nechung's words, he would have followed Nechung's advice in 1959 to stay in Tibet during the Chinese Invasion. Instead, he followed Dorje Shugden's, which told him to leave now - and thereafter, described the way out of Tibet. This was only in addition to Dorje Shugden's previous instruction of forming the guerrilla group, Chushi Gangdruk which safeguarded the Dalai Lama onto Indian territory.
Nechung's track record of divination's hasn't been too accurate either. Not only did Nechung almost 'cost' the Dalai's Lama's life, but his continuous prophesy of how Tibetan's would see a speedy return to a free Tibet, has yet to come true.
Then, there comes the blame game.
So much drama. But without drama, can the Dharma grow in this day and age?