  The heliocentric redshifts ($z_\mathrm{hel}$) reported for 150 Type Ia
supernovae in the Pantheon compilation are significantly discrepant from their
corresponding values in the JLA compilation. Both catalogues include
corrections to the redshifts and magnitudes of the supernovae to account for
the motion of the heliocentric frame relative to the `CMB rest frame', as well
as corrections for the directionally coherent bulk motion of local galaxies
with respect to this frame. The latter is done employing modelling of peculiar
velocities which assume the $\Lambda$CDM cosmological model but nevertheless
provide evidence for residual bulk flows which are discordant with this model
(implying that the observed Universe is in fact anisotropic). Until recently
such peculiar velocity corrections in the Pantheon catalogue were made at
redshifts exceeding 0.2 although there is no data on which to base such
corrections. We study the impact of these vexed issues on the 4.4 $\sigma$
discrepancy between the Hubble constant of $H_0 = 67.4 \pm 0.5$ km/s/Mpc
inferred from observations of CMB anisotropies by Planck assuming $\Lambda$CDM,
and the measurement of $H_0 = 73.5 \pm 1.4$ km/s/Mpc by the SH0ES project which
extended the local distance ladder using Type Ia supernovae. Using the same
methodology as the latter study we find that for supernovae whose redshifts are
discrepant between Pantheon and JLA with $\Delta z_\mathrm{hel} > 0.0025$, the
Pantheon redshifts favour $H_0 \simeq 72$ km/s/Mpc, while the JLA redshifts
favour $H_0 \simeq 68$ km/s/Mpc. Thus the discrepancies between SNe Ia datasets
are sufficient to undermine the claimed `Hubble tension'. We further note the
systematic variation of $H_0$ by $\sim$ 6-9 km/s/Mpc across the sky seen in
multiple datasets, implying that it cannot be measured locally to better than
$\sim$ 10% in a model-independent manner.
