What happened to Gandalf the Grey's body when he was sent back as Gandalf the White?

So Gandalf and the Balrog of Moria have a little fisticuffs and they both died. Presumably dead bodies lay on the mountain side somewhere.

Gandalf "strayed out of thought and time", gets a can of spinach and a new colour then he comes back sometime after his death as Gandalf 2.0.

The question is thus:

1. Did Gandalf old body get burnt to a crisp or some other death that mean there was no dead body left? OR


2. Did his old body get renewed?

Or do we not know? If his body was left in the snow then wouldn't it have been preserved frozen? This seems like an obvious oversight in the story.

I'd like to stick with the book canon but out-of universe answers are also welcome.