  The early dark energy (EDE) extension to $\Lambda$CDM has been proposed as a
candidate scenario to resolve the "Hubble tension". We present new constraints
on the EDE model by incorporating new data from the Dark Energy Spectroscopic
Instrument (DESI) Baryon Acoustic Oscillation (BAO) survey and CMB lensing
measurements from the Atacama Cosmology Telescope (ACT) DR6 and \textit{Planck}
NPIPE data. We do not find evidence for EDE. The maximum fractional
contribution of EDE to the total energy density is $f_\mathrm{EDE}< 0.091 \;
(95\% \; \mathrm{CL} )$ from our baseline combination of \textit{Planck} CMB,
CMB lensing, and DESI BAO. Our strongest constraints on EDE come from the
combination of \textit{Planck} CMB and CMB lensing alone, yielding
$f_\mathrm{EDE}< 0.070 \; (95\% \; \mathrm{CL} )$. We also explore extensions
of $\Lambda$CDM beyond the EDE parameters by treating the total neutrino mass
as a free parameter, finding $\sum m_\nu < 0.096 \,\, {\rm eV} \; (95\% \;
\mathrm{CL} )$ and $f_\mathrm{EDE}< 0.087 \; (95\% \; \mathrm{CL} )$. For the
first time in EDE analyses, we perform Bayesian parameter estimation using
neural network emulators of cosmological observables, which are on the order of
a hundred times faster than full Boltzmann solutions.
