  We focus on viable $f(T)$ teleparallel cosmological models, namely power law,
exponential and square-root exponential, carrying out a detailed study of their
evolution at all scales. Indeed, these models were extensively analysed in the
light of late time measurements, while it is possible to find only upper limits
looking at the very early time behavior, i.e. satisfying the Big Bang
Nucleosynthesis (BBN) data on primordial abundance of ${}^4He$. Starting from
these indications, we perform our analysis considering both background and
linear perturbations evolution and constrain, beyond the standard six
cosmological parameters, the free parameters of $f(T)$ models in both cases
whether the BBN consistency relation is considered or not. We use a combination
of Cosmic Microwave Background, Baryon Acoustic Oscillation, Supernovae Ia and
galaxy clustering measurements, and find that very narrow constraints on the
free parameters of specific $f(T)$ cosmology can be obtained, beyond any
previous precision. While no degeneration is found between the helium fraction,
$Y_P$, and the free parameter of $f(T)$, we note that these models constrain
the current Hubble parameter, $H_0$, {higher extent than the standard model}
one, fully compatible with the Riess et al. measurement in the case of power
law $f(T)$ model. Moreover, the free parameters are constrained at non-zero
values in more than 3-$\sigma$, showing a preference of the observations for
extended gravity models.
