We use the three-scale framework of Hu et al. to show how the Cosmic Microwave Background anisotropy spectrum depends on the fundamental constants. As expected, the spectrum depends only on emph{dimensionless} combinations of the constants, and we emphasize the points that make this generally true for cosmological observables. Our analysis suggests that the CMB spectrum shape is mostly determined by $alpha^2m_e/m_p$ and the proton-CDM-particle mass ratio, $m_p/mchi$. The distance to the last-scattering surface depends on $Gm_pmchi/hbar c$, so published CMB observational limits on time variations of the constants implicitly assume the time-independence of this quantity, as well as assuming a flat-lcdm~cosmological model. On the other hand, low-redshift BAO, $H_0$ and baryon-mass-fraction measurements can be combined with the emph{shape} of the CMB spectrum to give information that is largely independent of these assumptions. In particular we show that the pre-recombination values of $Gmchi^2/hbar c$, $m_p/mchi$ and $alpha^2m_e/m_p$ are equal to their present values at a precision of $sim15%$.
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