We present a detailed analysis of the impact of $H_0$ priors from recent surveys in the literature on the late time cosmology of five $f(T)$ cosmological models using cosmic chronometers, the Pantheon data set, and baryonic acoustic oscillation data. In this work, we use three recently reported values of $H_0$ that have contributed to the recent $H_0$ tension problem. We find that these priors have a strong response in these analyses in terms of all the cosmological parameters. In general, our analyses gives much higher values of $H_0$ when considered against equivalent analyses without priors while, by and large, giving lower values of the matter density parameter. We close with a cross-analysis of each of our model, data set and prior combination choices.
Comment: 24 pages, 11 figures