Charcoal, 122122 120 bytes

≔＆θ⁰η≔⟦η⟧ζＦζＦιＦ⁼κ⁺⟦⟧κ«ＵＭκ⊖μ⊞ζκ»⊞υθ⊞υηＦυＦιＦ⁼κ⁺⟦⟧κ⊞υκＷ∧υ⊟υ«≔⟦⟧ζＷι⊞ζ⊟ιＦ⮌ζ¿⁺⟦⟧κＦκ⊞ιλ⊞ικ»≔⮌θθ≔⁰ζＦ⁻η⊖⌊η«Ｆ⁻ζι]→Ｆ⁻ιζ[Ｉ⊟θ≔ιζ»Ｆζ]ＵＢ≔＆θ⁰η≔⟦η⟧ζＦζＦιＦ⁼κ⁺⟦⟧κ«ＵＭκ⊖μ⊞ζκ»⊞υθ⊞υηＦυＦιＦ⁼κ⁺⟦⟧κ⊞υκＦ⮌υ«≔⟦⟧ζＷι⊞ζ⊟ιＦ⮌ζ¿⁺⟦⟧κＦκ⊞ιλ⊞ικ»≔⮌θθ≔⁰ζＦ⁻η⊖⌊η«Ｆ⁻ζι]→Ｆ⁻ιζ[Ｉ⊟θ≔ιζ»Ｆζ]ＵＢ, 

Try it online!Try it online! Link is to verbose version of code. Explanation: Not an easy problem to solve without access to recursion.

≔＆θ⁰η 

Make a deep clone of the input and set all of the integers in that clone to zero. (BitwiseAnd fully vectorises, unlike, say, Times.)

≔⟦η⟧ζＦζＦιＦ⁼κ⁺⟦⟧κ«ＵＭκ⊖μ⊞ζκ» 

Perform a breadth-first search over the clone and decrement each sublist. (Again, it is fortunate that Decremented fully vectorises, although I still have to MapCommand it because I need to modify each sublist in-place.) The resulting list ends up with values appropriate to their depth.

⊞υθ⊞υηＦυＦιＦ⁼κ⁺⟦⟧κ⊞υκＷ∧υ⊟υ«≔⟦⟧ζＷι⊞ζ⊟ιＦ⮌ζ¿⁺⟦⟧κＦκ⊞ιλ⊞ικ»⊞υθ⊞υηＦυＦιＦ⁼κ⁺⟦⟧κ⊞υκＦ⮌υ«≔⟦⟧ζＷι⊞ζ⊟ιＦ⮌ζ¿⁺⟦⟧κＦκ⊞ιλ⊞ικ» 

Flatten both the input and the depth clone in-place.

≔⮌θθ≔⁰ζＦ⁻η⊖⌊η«Ｆ⁻ζι]→Ｆ⁻ιζ[Ｉ⊟θ≔ιζ»Ｆζ]ＵＢ, 

Adjust the depth list so that the minimum depth is 1, and use that to build the desired output list from the depth list and the flattened input list. (See my answer to Build a list from a depth map for more detail.)