[Mänwe]

To add to Alcuin’s excellent post I believe that A_Brandybuck is referring to the following note;

“There were and always remained some Dwarves on the eastern side of Ered Lindon, 3 where the very ancient mansions of Nogrod and Belegost had been – not far from Nenuial; but they had transferred most of their strength to Khazad-dûm.” – (Unfinished Tales, Concerning Galadriel and Celeborn.)

Though I do find the reference to the Nenuial rather strange, “not far from” when clearly the lake (I think you made a typo there A_Brandybuck) is a great distance from the Ered Luin, situated against the Hills of Evendim and consequently Belegost and Nogrod.

Alcuin provides us with evidence from “The War of the Jewels” that Belegost is north of Mount Dolmed, which does not contradict what is mentioned in the Silmarillion;

“To the north of the great height of Mount Dolmed was Gabilgathol, which the Elves interpreted in their tongue Belegost, that is Mickleburg; and southward was delved Tumunzahar, by the Elves named Nogrod, the Hollowbold.” – (The Silmarillion, Chapter 10, Of the Sindar)

I do not think that the distance is explicitly stipulated, I cannot help quote Christopher’s words on the matter;

“Whether Belegost was near to or far from Nogrod is not made plain;…” – (The Book of Lost Tales Part II)

Judging the distance by eye from the various maps will have to suffice..

As to whether they were destroyed, I believe that both were. If we take the Appendix of Lord of the Rings as Tolkien’s ‘true’ beliefs then;

“After the end of the First Age the power and wealth of Khazad-dûm was much increased; for it was enriched by many people and much lore and craft when the ancient cities of Nogrod and Belegost in the Blue Mountains were ruined at the breaking of Thangorodrim.” – (Appendix A, III Durin’s Folk)

I would suggest is conclusive.