I would assume that the labeling occurs in the reduction of NTPs to form dNTPS.
This process (catalyzed by ribonucleotide reductase) involves protonating the hydroxyl group on the 2' carbon, allowing it to leave as water, and then adding a hydride to the newly formed carbocation. The two hydrogen atoms (the proton and the hydride) come from two thiols on the enzyme, which in the process are oxidized to form a disulfide bond.
The crux is that, in the presence of heavy water, the two thiols would rapidly exchange their hydrogen for deuterium. That means that the hydride that gets added to the carbocation would be a deuterium, and the resulting dNTP would be deuterium labeled.
This is the only step that I can imagine the labeling working for, as a carbon-hydrogen bond is formed (which doesn't exchange rapidly with the solvent) using the hydrogen from a sulfer-hydrogen bond (which does exchange rapidly with the solvent).