By Thrain and Thorin, I assume you are refering to Thrain II and Thorin II, Samwisegamgee, who for many years spent part of their lives exciled in the Blue Mountains.
I have found very little information about this subject. Like Grondmaster says earlier, Nogrod would lie beneath the Gulf of Lune, so there is no possibility of them living there. Belegost was said to have been destroyed too, but it was a lot further south than Nogrod in the area that became Harlindon. It is possible that industrious Dwarves could have mined their way into the ruins, but personally I don't believe that was the case.
Thorin was said to have done alright for himself but only through hard work. I got the impression he had spent those years slaving over a forge and anvil working iron, and that was why he was so keen to take back Erebor. Likewise with Thrain wanting to recolonise Khazad-dum. Belegost was supposed to have been rich in precious metals and gems. I think if they were living there, they would have been content to stay there, and would have been proud to say they came from Belegost.
One of the only other mentions of mines in these mountains is a story about King Arvedui, who in TA 1974 took shelter in an abandoned Dwarf mine in the northern Ered Luin. Since the Kings of Durin's Line were living there between TA 2810 to 2941, ie approx 900 years later, it is possible they had recolonised these same mines in the North.
As a sidenote, Thrain I lived in Khazad-dum until he led a group of his people to Erebor in TA 1999, and Thorin I lived in Erebor until leading his people into the Ered Mithrin .