No, they don't mix. Water molecules are polar; they have charges, positive on one end, negative on the other. So, water molecules attract water molecules or any other polar molecule). Oil molecules are non-polar. They get sort of frozen out, like the cool kids hanging together and ignoring the nerds. Happily for gin and tonic fans, alcohol is also polar and mixes nicely with tonic water.
But oil and water do emulsify, and mustard is not required. Mustard is just ground seeds and vinegar, and the liquid component of mayo is already vinegar. They will emulsify with just oil and water, meaning oil and vinegar, but, as with scratch oil and vinegar salad dressing, it's not stable. The egg yolk is the emulsifier, providing stability, because it provides long amino acid chains, proteins, that have the ability to link fat and water. (Lecithin is another emulsifier, but a different kind. It is a surfactant that attaches to fat at one end and water at the other. You see it a lot in manufactured foods.) Kraft Mayo also contains food starch as a stabilizer. A stabilizer has a large molecule that simply keeps fat particles away from each other so they can't combine. Viscosity also stabilizes, because it's harder for the fat particles to move against each other, and the mustard that Kraft uses in its mayo has some fruit pectin that helps thicken.
But we don't often use stabilizers. We could, but I don't think most who care enough to make their own make large batches and have to worry about them breaking down. And, although we do usually add mustard, most prepared mustards are vinegar, seeds, water, salt and spices. No pectin or other thickeners and certainly no emulsifiers.