Will (from Barrister and Mann) analyzed the MWF scent a few years back, as he did with many other (mostly perfume) scents. Here it is copy/pasted:
Nope, definitely not a fougère.
It's primarily phenethyl alcohol (which is very powerful and likely not included at a high enough level for it to be part of their allergen declaration, but I definitely smell it), ionones, and trace amounts of various citrus-related compounds like hydroxycitronellol. It does also contain linalool (major component of lavender oil), but so do tons of other things that aren't fougères. Linalool is an extremely common ingredient in fragrances as a whole, so I generally ignore it as being indicative of anything in particular.
They list the scent profile as "Lavender, Pine and Clove on a spicy base of Nutmeg, Cedarwood, Sandalwood, Vanilla and Musk." Let's break apart the compounds listed in the allergen declaration to see where they get each of those. Some probably won't show up because they're either damn powerful and thus need not actually be included in the ingredient list because they're used at such low levels (like vanillin, which, if used in any great quantity, will heavily discolor the soap). Keep in mind that it's been a bit since I smelled Mitchell's Wool Fat, so I'm theorizing here what each particular declared ingredient is meant to represent, along with what I remember of the fragrance.
Lavender
Linalool. LOTS of it. Probably also some linalyl acetate, which is another major component of lavender oil, but is likely used here at such a low level that it doesn't cross the allergen declaration threshold (wash-off shaving products fall into IFRA category 9a, so the total fragrance concentration is not to exceed 5%, but I cannot find a resource for individual allergen declaration thresholds).
Pine
Limonene (which, along with terpineol, is a major component of pine oil in addition to multiple citrus oils) and hydroxycitronellal (an extremely versatile compound that occurs naturally in quite a few flowers).
Clove
Hexyl Cinnamal, also called Hexyl Cinnamaldehyde. Has a cinnamon-like scent at full concentration but, in dilution, can smell like cinnamon, clove, nutmeg, or any number of other spices. My guess is that they use it as the major "spice" note and then add minor corollary notes to emphasize specific facets.
Nutmeg
See my reference to clove.
Cedarwood
Not sure, to be honest, mostly because there are tons of woody synthetics out there. My guess is that they're incorporating methyl cedryl ketone, palisandin, iso e super, or some combination of those, all of which are cheap and fairly powerful, to generate the cedar and patchouli notes. You'd be surprised how much overlap there is.
Sandalwood
Again, they're probably using something cheap and POWERFUL; my guess would be the aromachemical indisan, which you can find by its chemical name iso camphanyl cyclohexanol. It goes for about nine cents per gram on the open market, but bulk purchases drive the cost down. I'd be surprised if they pay more than $30 USD per kilo for the stuff.
Vanilla
Most things that smell like vanilla will also oxidize/discolor soap like vanilla, which leads me to believe that they're either using very little of it (makes sense; I don't remember the fragrance being tremendously vanillic) or using something like heliotropyl alcohol, which smells very much like vanillin, is quite powerful, and won't discolor soap, but is also a weird/rare molecule and thus not cheap. My money's on the former.
Musk
Something cheap and powerful. They would have used nitromusks when MWF was first created, but nitromusks have been banned for 50 years. I'd guess low levels of ethylene brassylate and probably a macrocyclic musk that smells a bit like nitromusks would (possibly galaxolide, which is actually a polycyclic musk, but is somewhat possessed of the velvety character for which nitromusks were so famous). Still, most of the powderiness is probably due to the ionone content, so I doubt there's all that much musk in it.
Patchouli (because I've seen it mentioned in a bunch of the note descriptions for MWF/Kent)
Probably created by use of the phenethyl alcohol I mentioned along with α-isomethyl ionone, which also imparts a powdery, clean scent that would work well with the musks that I discussed preciously. Most ionones have facets of almond, violet, iris, and raisin (particularly noticeable in methyl ionone ß) at full strength, but the family as a whole is tremendously adaptable and can be used to impart any number of particular fragrance facets.
