This is basically a poorly formulated lotion. The rules for lotion apply, not the rules for soap.
I want to warn about the hazards of blending water based ingredients (the aloe) with fats or fatty acids (the shea or wax). You really need a preservative in there because the mix of water, carbs, and fat are a great buffet for microbes.
And an emulsifier would be a good idea too. All the shea (or wax) is doing is adding thickening to a mechanically blended mixture. There's no emulsification, which adds chemical stability to ensure the mixture stays mixed. If you don't also emulsify, the mixture is not stable and may separate.
Craig has a valid point. In this type of product, the properties of the wax or shea are the only properties to concern yourself with. What fatty acids are in the wax or the shea are meaningless. You aren't breaking these ingredients apart as you do in saponification.
Wax is sometimes used in lotion making, but it wouldn't normally be used in such a large percentage of the hydrophobic portion (the fats, oils, etc.) I suspect the texture would be really stiff and unworkable. You really need something with a softer texture and a melt point nearer normal skin temp -- like the shea or another butter.