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I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more complicated operators, like the Hamiltonians of the Hydrogen atom or harmonic oscillator?

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The Wigner-Weyl transform of a function $f(x,p)$ is given by,

$$\Phi[f]= \frac{1}{4\pi^2}\iiiint f(x,p) \exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db$$

As you suggested, let us take the Hamiltonian of the harmonic oscillator, i.e.

$$\Phi[H]=\frac{1}{8m\pi^2}\iiiint p^2\exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db \\ + \frac{m\omega^2}{8\pi^2}\iiiint x^2\exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db$$

We concern ourselves with the last integral, as they are more or less analogous. As the first integration is over $x$, we may applying integration by parts and ignore $p$:

$$\frac{m\omega^2}{8\pi^2}\iiint \frac{1}{a^3}e^{ia(X-x)+ib(P-p)}(ia^2 + 2ax-2i) \, dp \,da \, db$$

Integrating with respect to $p$ is trivial:

$$\frac{m\omega^2}{8\pi^2}\iint \frac{b}{a^3} e^{ia(X-x)+ib(P-p)}(ax^2 +2-i2ax) \, da \, db$$

With the help of Mathematica 9, we may express the subsequent integral over $a$ in terms of a polyanomial, and the exponential integral function:

$$\frac{m\omega^2}{8\pi^2}\int \, b e^{ib(P-p)} \, \left[ (X-x)((ix^2+4x)-2X)\mathrm{Ei}(ia(X-x)) \\ -\frac{1}{a}e^{ia(X-x)}(i(X-x)+(x^2-i2x)+1) \right] \, db$$

The integral over $b$ is also trivial, as the integrand only features $b$ in the form $be^{b\dots}$ Hence,

$$-\frac{m\omega^2}{8\pi^2(P-p)^2} \left[ (X-x)((ix^2+4x)-2X)\mathrm{Ei}(ia(X-x)) -\frac{1}{a}e^{ia(X-x)}(i(X-x)+(x^2-i2x)+1) \right]e^{ia(X-x)+ib(P-p)}\left( ib(P-p)-1\right)$$

Apply the same procedure to the original first integral, combine the two, etc.

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The standard name for what you are seeking, is the Wigner transform, the inverse of the Weyl transform. (As the Weyl transform maps phase-space functions to operators.) for an arbitrary operator in any ordering, the Wigner transform follows a simple 1964 formula by Kubo, eqn (111) of Ref. 1, effectively the Fourier transform of the off-diagonal matrix elements of said operator between position eigenstates.

The Wigner transform of the Coulomb potential is well-known to be an unyieldy integral expression (there are better ways for solving the Hydrogen atom in phase space). For the oscillator Hamiltonian, it is the standard expression, in normalized undimensionalized units, the square of the radius in phase space, $(p^2+x^2)/2$. For the typical operator expression exp(ax̂) exp(bp̂), the Wigner transform is, as per that formula, exp($ax+bp+i\hbar ab/2$).

A more celebrated Wigner transform is that for the evolution operator of the oscillator, $\exp(\frac{it}{2\hbar} (\hat{x}^2 + \hat{p}^2) )$, namely eqn (60) of Ref. 1, $$ \frac{1}{\cos (t/2)} e^{\frac{i\tan(t/2)}{\hbar} (x^2+p^2)} ~. $$

References:

  1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.
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