The key word is "mixing". If you take a glass cylinder and fill it halfway with D2O (the heavier water, so put it on the bottom) and fill the cylinder ever so gently with H2O, the mixing will be determined by diffusion (NO mechanical mixing). But the experimental techniques required to analyze and determine the rates would be expensive and time-consuming.
Now if in a practical situation you need to make HOD, you could fill a beaker halfway with one and the rest of the way with the other (doesn't matter which way), and stir with a glass rod or a high speed mixer - or shake the mix in a bottle. The time to make HOD will be as fast as you can mix by shaking or stirring.
However, the idea of watching a diffusion experiment sounds like a fun thing. A cylinder could be filled halfway with dyed water, then covered with pure water, and the color equalization could be observed. It should probably be done with pure water on the bottom and dyed water on top, too, for comparison. And probably with more than one dye (in separate experiments) to see if the dye is important in the diffusion rate, or whether the dye is just carried by larger bunches of water.
The exchange of protons in amines or between H2O and other hydroxylic compounds has been found to be measurably slow (e.g., https://pubs.acs.org/doi/abs/10.1021/ja01652a030, which dates to 1953). A longer article in Wikipedia (https://en.wikipedia.org/wiki/Hydrogen–deuterium_exchange) discusses proteins and other compounds that are separated from the H2O/D2O environment and analyzed separately. What I am suggesting (and may be on shaky ground) is that H2O and D2O in solution together are not as independent as molecules of proteins or alcohols. In H2O, all the hydrogens are equivalent and exchange on a time scale that might be calculable; same for D2O. But when you put bulk H2O and bulk D2O together, the hydrogen bonded clusters will re-form, rearrange, simply by being there, so that if you try to analyze for HOD, you are selecting one molecule (say, by evaporation) from a cluster that is primarily determined by how much mixing you have done. If you add a drop of H2O to a liter of D2O, you may be quick enough to recapture most of that drop. But if you mix the drop (takes time: maybe one or two seconds), those H2O molecules are distributed throughout the liter, forced into new hydrogen bonding by the mixing, and not identifiable as H2O because they are in clusters of molecules. Pull out an oxygen, and it might have an H and a D, or 2D, but not likely 2H, because of the randomness induced by the mixing.