Way and Nussbaumer (2011, arXiv:1104:3031): "Two years later Hubble found the same velocity-distance relationship on observational grounds ... from practically the same observations that Lemaitre had used."
Van den Bergh (2011, arXiv:1108:0708): "1929 Hubble repeats Lemaitre's work with essentially the same data and obtains similar results"
Although these assertions sound plausible, they are actually quite wrong - while Lemaitre and Hubble did use the same published velocities and both relied on Hubble's distances to nearby, Local Group galaxies, each used a different method and different data to derive distances to more remote galaxies. Furthermore, each made a serious blunder in the analysis that greatly impacted their determination of the Hubble constant, and the fact that both arrived at a similar value is actually sheer accident. These blunders are of interest since they are part of the larger story as to why the early values of the Hubble constant were so much bigger than modern determinations (~70 km/s/Mpc).
As an initial matter, we note that Hubble's distances to the nearby Local Group galaxies were all too small. Roughly, two effects were at work. First, the calibration of the Cepheid period-luminosity relation (from Shapley) was too faint by about 1.5 magnitudes. Second, the Mount Wilson stellar magnitude scale had a nonlinearity between the bright and faint ends. The impact of this nonlinearity is difficult to assess but might have added another 0.6 mag error to the distance modulus. These errors affected both Lemaitre and Hubble - had these been the only errors, both would have computed a value for the Hubble constant of around 200 km/s/Mpc, much closer to the modern value.
Lemaitre relied on Hubble's Eq. 8, which presumes that the mean absolute magnitude of nearby galaxies can be applied to more distant galaxies as well. However, Hubble couched this result with caveats like "extrapolation", "assumption", and "working hypothesis", implying that it was a conjecture that needed to be validated later. Lemaitre, however, ignored these caveats and adopted the equation as established fact. We now know that Hubble's conjecture was wrong. Today we would call the problem one of Malmquist bias - a sample of nearby galaxies will be dominated by the numerous, intrinsically faint objects, while at large distances, due to selection effects, one only sees (and measures) the rare but nevertheless intrinsically bright galaxies, introducing a bias. Indeed, Lemaitre's sample of more distant galaxies was intrinsically brighter by 2.3 mag in comparison to Hubble's calibrating sample of nearby galaxies. This effect accounted for Lemaitre's erroneously high value of the Hubble constant.
Hubble's brightest star method, while not very precise, was not subject to the biases encountered by Lemaitre. Further, there is a physical basis (the Eddington limit) as to why the method should work. Hubble's problem, however, was that he was unable to distinguish individual stars from HII regions in the more distant galaxies. Humason, Mayall, and Sandage (1956, AJ, 61, 97) would later show that Hubble's "brightest stars" for his most distant galaxy in the Virgo cluster were, in fact, just such HII regions, and that the brightest stars were 2 magnitudes fainter. This effect accounted for Hubble's erroneously high value. The error was nearly the same size as Lemaitre's, and also accounted for why Hubble, in a further erroneous bit of reverse engineering, concluded that his conjecture about the universality of a galaxy's absolute magnitude appeared to be correct ("... this entirely unforced agreement ...").
Notwithstanding the suggestive straight line drawn in the left figure, Lemaitre had to assume that the linear relation, passing through the origin, is correct ab initio. He could not infer it from the data - the scatter was too large. Lemaitre was well aware of this problem and attributed it (correctly) to the fact that, as Hubble had shown, there is a large scatter in the intrinsic absolute magnitude of galaxies about the mean, and this scatter is nearly as large as the effect he was looking for, given the limited range in velocity of galaxies that was available to him. This explanation also accounts for why previous researchers, such as Lundmark and Stromberg, were unable to discern the relation from very nearly the same data. Conversely, even though Hubble had a much smaller sample of galaxies than Lemaitre and his differentiation of stars from HII regions was imperfect, his ordering of the galaxy distances was much better, and thus he was able to demonstrate the existence of a linear velocity-distance relation, passing through the origin, based purely on the data alone.
We can make this comparison more quantitative. A Spearman rank-correlation coefficient is a nonparameteric measure of the degree and significance of a monotonic correlation between two variables. The data for 42 galaxies in the Lemaitre diagram, on the left side above, have a correlation coefficient of 0.39 with a significance of 2.5 sigma - low enough that the data are only marginally inconsistent with there being NO correlation at all. The data in the Hubble diagram, on the right side, even though consisting of only 24 galaxies, have a correlation coefficient that is much higher - 0.85 - with a significance that is also greater - 5.5 sigma. If we trust in the mythical 3 sigma as being the threshold for making a discovery, credit for discovery of the velocity-distance relationship goes to Hubble.