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Although the experimental tests were not significant,
can we conclude anything from the correlational
results?

The correlations suggest that the variables are related to each other in individuals (e.g. if an individual has an above-average score on "Distancing", then they will tend to also have an above average score on "Dangerous").

Such correlations don't imply that there would be differences on any individual score between two groups (Remain objective vs Perspective taking); there might be but there might not be -- neither precludes such a correlation (though a substantial difference between groups on these measures would impact a correlation calculation that ignored this difference).

Here's an illustration; the left plot shows a plot of two correlated variables where there's no difference in group means (groups indicated in blue vs reddish-brown). The second plot
shows two correlated variables with a difference in group means on each variable but the differences are opposite in sign; this induces a small
negative correlation if you ignore the group factor.

As we can see ignoring
the grouping variable would lead to a mistaken impression of how the two variables are related.

If the mean-shifts had been in the same direction on the two variables, the induced marginal correlation between the x and y variables would have been positive even though the variables might not have been related within groups:

can we conclude anything from the correlational results?

Yes, it seems that some of those measures are correlated with each other.

[However, we should carefully consider a caveat: if there's some variable or variables -- possibly ones on which we have no data -- which does cause a difference in means, that might in turn induce the correlations that are observed, just as with the last plot above. Whether that's an issue depends on what you're trying to do.]