Mostly by reaction ratios, like the stoichiometry (the thing where they reacted 2 ‘volumes’ of hydrogen to 1 volume of oxygen to make water) but there’s several ways to measure atomic mass.
I think we really started measuring it analytically with Millikan’s experiments with oil droplets though. He measured the charge of an electron with high precision.
From there you can calculate by deflecting oil droplets of a known substance. Different masses will deflect differently under the same charge. This is essentially how mass spectrometers work.
From there you can go from known masses to how many protons and neutrons it has by the amount it deflects.
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