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In this work we show that the cosmological lithium problem and the $H_0$tension could be eased at the same time by allowing variations in thefundamental constants. We compute the primordial abundances of light elementsresulting from Big Bang Nucleosynthesis considering the fine structureconstant, the Higgs' vacuum expectation value and Newton's constant as freeparameters. Using the observational data for abundances, we set constraints onthe variations of the fundamental constants. An interpretation of the resultsin terms of the number of effective relativistic species gives $N_{\rmeff}=4.04\pm0.12$. If one extrapolates the fit of {\sffamily Planck}considering this $N_{\rm eff}$, the value of the inferred Hubble's constantshifts to $H_0=(71.85\pm0.77)\,\text{km}\,\text{s}^{-1} \text{Mpc}^{-1}$,compatible with current direct determinations.

The cosmological lithium problem, varying constants and the $H_0$ tension