How statistically significant was the "sweaty T-shirt" experiment?
First, let's consider the original Wedekind et al (1995) study.

Sample sizes seem reasonable, if the effect is not very noisy:


  [...] 49 female students (average age: 25.2 years, s.d. = ­4.0) and 44 male students (average age: 24.7 years, s.d. = 2.6) [...]


... and the statistical design accounts for individual differences in a fairly robust way:


  [...] women were asked to rate the odours of six T-shirts each, three of them worn by men who were dissimilar to the rating woman's MHC (average number of dissimilar HLA-antigens = 5.9, s.d. = 0.26), and three worn by men who were more similar to it (average number of dissimilar HLA-antigens = 2.7, s.d. = 0.74). We tried to present every T-shirt as often to MHC-dissimilar women as to MHC-similar women (average difference of presentations to thc two groups: -0.02, s.d. = 0.73). The presentation was random in every other respect [...]


Rather irritatingly, although the study design involved asking women to rate at least three attributes of the smells (sexiness, pleasantness, and intensity) the study performs a bait-and-switch in its results:


  The scores for sexiness are not shown in the figures as they were highly correlated with pleasantness (all scorings: r = 0.85, n =  294; for women who do not take the pill: r = 0.87, n = 186; for women who take the pill : r = 0.83, n = 108, p always &lt;&lt; 0.001)


This is a bait-and-switch because even if A correlates with C, and B correlates with C, we do not know that A and B correlate with each other. Of the three measured attributes, 'sexiness' arguably has the most direct bearing on mate-choice. Because analysis for 'sexiness' is not presented, the paper ends up telling us much less than it could have about the relationship between MHC relatedness and mate choice.

The results for 'pleasantness' and 'intensity' look like this:



One further piece of analysis is presented: some number of the female subjects seem to have reported that particular smells reminded them of their partner or ex-partner; or reminded them of family members. Mentioning that a shirt smelled like a partner or ex-partner happened more frequently for MHC-dissimilar shirt smells (~17% vs ~9%; p = 0.038, two-tailed Fisher exact test). Oddly, no written response-collection is described in the methods, so I can only assume that these assessments were ad-hoc.

If this was all of the analysis that had been done on the topic, my assessment would be: this is amongst the many published research findings that are probably false. My main concerns would be that 


Not all of the collected data are presented;
The most directly-informative contrast is not shown;
No adjustments for multiple comparisons are made on the statistical tests;
The data that are shown are very noisy, and sample sizes are not huge, raising concerns that the tests are underpowered;
The authors conduct tests based on data-collection which is not detailed in their methods, raising concerns about p-hacking.


And now, the 2005 paper!

Right off the bat, this paper cannot possibly replicate the findings of the Wedekind (1995) paper, because Wedekind et al found significant preferences for dissimilar MHC odours in women not taking the pill, and significant preferences for similar MHC odours in women taking the pill. In this study:


  [...] we did not distinguish between, in our statistical analysis, women who were taking oral contraceptives from women who were not.
  
  [...] Nine female participants (31.03% of female subjects) were taking oral contraceptives.


Sample sizes are smaller than Wedekind 1995 in terms of number of participants, but larger in terms of number of contrasts, because this experiment had two different sample types (sweat odour and urine), and had men smelling women as well as women smelling men. Within-individual effects are not modelled, so there is a potential for (e.g.) a small number of generically good-smelling individuals with rare MHC alleles at the population level to bias the analyses - whereas this potential bias was accounted for in Wedekind by performing within-individual correlations. The sample-sizes look like this:



The analysis was a series of Chi-squared tests, for data collected in four different sessions: one with men smelling female urine, one with men smelling female body odour, one with women smelling male urine, and one with women smelling male body odour. All chi-squared tests returned non-significant results with the exception of women smelling male body odour, for which the Chi-squared table looks like this:



So, this study didn't find the same result as the Wedekind study, but then, it couldn't find the same thing as the Wedekind study because of the way the analysis was performed, so that is no surprise. The authors claim that they found a significant effect of MHC relatedness on odour perception, but only in the proportions in the hard-to-interpret 'indifferent' response category. I am inclined to disagree. The authors presumably conducted 16 contrasts (a test of independence and three goodness-of-fit tests in each of four treatment types), and found one instance of p &lt; 0.05. I would argue that the one 'significant' result is best explained as a failure to account for multiple comparisons.

Ending notes

If these two papers were the only things that had been published on MHC-directed mate preferences in humans, I would be far from convinced that the claimed effects were real.

however...

These are far from the only papers on the subject of MHC-driven mate-choice in humans. Wedekind and Füri (1997) conducted an analysis with very similar experimental design to Wedekind et al (1995), but arguably better statistical analysis, and found the same direction of association as Wedekind et al (1995), albeit still with a fairly high noise-to-signal ratio. Subjectively, this goes a reasonable distance towards making me think that the findings of Wedekind et al (1995) were not just a statistical fluke. I'm still not utterly convinced, but certainly more convinced than I would be if Wedekind's final paper on the topic had been the 1995 publication.

MHC-driven mate-choice has also been investigated using facial preferences and mate choice surveys. The combined evidence (across odour preference, facial preference, and mate choice surveys) is reviewed in Havlicek and Roberts (2009).