By adding the oil in slowly you are aiding in the homogenization of the oil droplets to smaller oil droplets with the high-shear mixing. The oil droplets get smaller in the mixing vortex by cavitation, and this maximizes surface area for the emulsifying proteins from the mustard to effectively coat the smaller oil droplets and stabilize them as an emulsion. Based on Stokes equation the smaller the oil droplets can be the slower they will separate to surface (also-- increasing viscosity of continuous phase (add xanthan), decreasing density difference of two phases (use 3% brominated vegetable oils), and decreasing effect of gravity (no centrifugation!) will all aid in stabilizing the emulsion).
By adding in oil all at once, the high-shear mixing is not as effective in breaking up the oil to smaller droplets (ideally you want 1-10 microns diameter) and now the proteins bind some oil but instead prefer forming aggregates with each other, no longer effective in coating the oil. Also the type of emulsifier used makes a big difference. A high shear-propeller system likely makes the least stable emulsions, a tissue homogenizer (Turrax blender) is better, next a high pressure piston-pump cavitation homogenizer is very good (most common in industry) while a microfluidizer would be considered the best emulsions with the smallest oil droplet sizes but these are more for small-scale lab applications and tend to get clogged often. It is useful to mention that your failed (separated) emulsion may be able to be saved by switching to a better homogenizer/emulsifier system where it would be inputting much more shear energy into system then your current mixer is.
Typically good emulsifying proteins have flexible conformations with an amphiphilic nature (charged hydrophilic amino acids on exterior with hydrophobic amino acids in core and with some properly placed cysteine residues but not too many!). This is why another trick in industry is to pre-mix at high shears using a Turrax homogenizer (at >10,000 rpm) the water system with mustard emulsifying proteins and soluble ingredients (no oil added yet) to help open up the protein’s conformation to aid in rapidity that these will be able to coat surface of oil droplets. Then oil is added slowly until all fully emulsified, and then for added stability this emulsion mix would be added to a piston-pump homogenizer and passed through several times. Measuring oil droplets with a Malvern Mastersizer would show oil droplet size distribution decreasing with each additional step but with greatest decrease going from Turrax to high pressure piston homogenizer.
This also explains why when making a salad dressing, the vinegar is typically the last ingredient added after the oil is effectively emulsified, since vinegar decreases pH and adds protons to negatively-charged amino acids making them more hydrophobic. By adding vinegar in before oil emulsification step, you have a more likely chance for same effect described above where emulsifying proteins become more protonated and more hydrophobic and prefer protein-protein interactions forming aggregates and thus when oil is added (even if added slowly) it may not be as effective in forming a nice stable film over the decreasing sized oil droplets with overall larger surface area, and your emulsion will be unstable and separate.