I've been looking to do this guy proper for the last 20 odd years now, and through lots of research of my own, if you want to achieve it the way they did it, you'll need a life case of your own face smiling. The overall prosthetic is composed of six pieces: the nose extension, the chin, two cheeks, and two mouth corners. They were sculpted and casted individually from molds and ran in foam latex. Because Nicholson is allergic to spirit gum, I believe they used a medical adhesive to stick it to his face.
The make up itself is something called PAX paint; it's a 50/50 mix of acrylic paint and pros-aide adhesive. This combo makes it flexible without cracking and lasts long enough to shoot with. The contrast/shadow areas are done with gun-metal gray powder and I believe some blue and reds above the eyes. The make up was then sealed and talc'ed so it wouldn't smear or stick if they needed to put the flesh tones over it (which was an oil based make-up stippled on top of his regular white make-up and he couldn't touch it once it was applied). To get it to run or rub off in-camera, the rag used to wipe his forehead was soaked in isopropyl alcohol and, most likely, a watered down jug of it was used to get it to run and smear when Vicki throws a jug of water on him in the museum scene (editing helped with the final look of it, I'm sure).
For the rest of the costume, you can source bits and bobs of it from many places. I have a pattern for vintage dress (evening tails and pants) that I used to make the pants. I have patterns for the pants and ties through my Spoonflower shop
here, if you want to give it a go. Indy Magnoli is the only person I know that offers the rest; shoes, waistcoat (of which I want to make accurate button closures for, in the near future) but get it with a purple satin back, if you're striving for accuracy; gloves, and I'm sure you can custom order the shirt. The suspenders, though briefly seen, is turquoise with a purple stripe running vertically. I asked Indy if he could make it for me but he recommended that it'd be cheaper to get a turquoise suspender and some purple fabric markers and do it yourself; I would recommend the same.
The rest is on you, friend. Good luck.