I just posted an answer to thisthis related MO question. To sum up the part that's relevant here:
Let $\mathcal N$ be the Baire space and let $X$ be any other zero-dimensional Polish space that is not $\sigma$-compact. Then there are continuous bijections $\mathcal N \to X$ and $X \to \mathcal N$.
I don't have a short proof of this assertion, but you can look at my paper for a proof, and at another paper of mine (joint with Arnie Miller) where some similar things are explored.
Now, let's fix a particularly nice $X$: say $X$ is the disjoint sum of $\mathcal N$ and the Cantor space $\mathcal C$. For this special case, I can give you a short proof that they're bijectively related (modulo a few well-known results).
To get a continuous bijection $\mathcal N \to X$:
By Exercise 7.15 in Kechris's Classical Descriptive Set Theory, a nonempty Polish space $Y$ is perfect if and only if there is a continuous bijection $\mathcal N \to Y$. $X$ meets these requirements.
To get a continuous bijection $X \to \mathcal N$:
There is a homeomorphic copy of $\mathcal C$, say $K$, with $K \subseteq \mathcal N$. By Theorem 7.7 in Kechris's book, $\mathcal N \setminus K$ is homeomorphic to $\mathcal N$. Thus we can get a continuous bijection $X \to \mathcal N$ by mapping $\mathcal C$ homeomorphically onto $K$ and mapping $\mathcal N$ homeomorphically onto $\mathcal N \setminus K$.
