After John Dalton used the law of multiple proportions to prove the existence of atoms, chemists began estimating the relative atomic weights of the elements using hydrogen as the basic unit. At what point did it become clear to scientists that all atomic weights were integer multiples of hydrogen's weight?
The premise of the question is untrue - atomic masses are not integer multiples of hydrogen.
After Dalton started measuring atomic masses circa 1803, scientists did quite quickly come to the conclusion that they were all exact multiples of hydrogen. This was known as the Prout hypothesis after William Prout, who published the idea in 1815 (here is an online copy of his original paper).
However, Prout's hypothesis was just as quickly dismissed by further measurements, mainly by Berzelius, that showed many elements do not fit the trend, e.g. chlorine with an atomic mass of 35.46 (in modern mass units). We now know that this is in part due to the presence of isotopes (chlorine is a mixture of 76% chlorine-35 and 24% chlorine-37) but even single isotopes do not have masses that are exact multiples of hydrogen's mass.
Why is this? Well, atomic nuclei are composed of protons and neutrons, and these have slightly different masses. A proton weighs about 1.007 mass units, a neutron weighs 1.009 mass units, and this discrepancy adds up, especially when you're dealing with heavy elements (e.g. the most common isotope of lead has 82 protons and 126 neutrons). Then you have the "mass defect" aka nuclear physics weirdness, which means that an atom actually weighs less than you would expect from just adding up the masses of the protons and neutrons in it. This was discovered by Einstein around 1905, and comes from the energy used to bind the nucleus together. Remember E=mc^2 ? Fusing protons and neutrons together to make atoms releases a huge amount of energy, which is equivalent to this mass defect multiplied by the speed of light squared.