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I have 2 point layers spread out all across the globe. I would like to use the Distance Matrix tool, and would like to get the distances in metres or kilometres. I'm aware that I have to change the projection to a UTM projection.

But since my data is across all longitudes, which projection should I use? I tried a few (EPSG 3857) to no avail; I still get the distances in decimal degrees.

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Depending on where your points are distributed on the globe (do you have points in the polar regions as well?), a global Mercator projection should do fine.

Did you reproject the map view in QGIS (on the fly) or the data itself?

You have to reproject the data (e.g. right click on the layer > Save as > Choose CRS). Only if both source data layers are transformed / reprojected and in the same CRS, the results are going to be valid.

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