  We use a combination of high-redshift observables to extract the strongest
constraints to date on the fraction of axion fuzzy dark matter (FDM) in the
mass window $10^{-26}\,\mathrm{eV}\!\lesssim\!
m_\mathrm{FDM}\!\lesssim\!10^{-23}\,\mathrm{eV}$. These observables include
ultraviolet luminosity functions (UVLFs) at redshifts $4-10$ measured by the
Hubble Space Telescope, a constraint on the neutral hydrogen fraction from
high-redshift quasar spectroscopy, the cosmic microwave background optical
depth to reionization measurement from Planck and upper bounds on the 21cm
power spectrum from HERA. In order to calculate these signals for FDM
cosmology, we use the 21cmFirstCLASS code to interface between AxiCLASS and
21cmFAST and consistently account for the full cosmic history from
recombination to reionization. To facilitate a full Bayesian likelihood
analysis, we developed a machine-learning based pipeline, which is both
accurate, and enables a swift statistical inference, orders of magnitude faster
than a brute force approach. We find that FDM of mass $m_\mathrm{FDM} \!=
\!10^{-23} \, \mathrm{eV}$ is bound to less than $16\%$ of the total dark
matter, where the constrains strengthen towards smaller masses, reaching down
to $1\%$ for $m_\mathrm{FDM}\! =\! 10^{-26} \, \mathrm{eV}$, both at $95\%$
confidence level. In addition, we forecast that a future detection of the 21cm
power spectrum with HERA will lower the upper bound at $m_\mathrm{FDM}\! =\!
10^{-23} \, \mathrm{eV}$ to $\lesssim\!1\%$.
