I haven't loaded full wadcutters in decades, but when I did I used whatever .38 brass I had at the time. If you were loading milspec .38 brass it may still be a heavier thicker wall case. The old white box Winchester brass was and may still be heavier. I never searched it out for wadcutters.
I've loaded .38 spl for 50 yrs, and recently loaded another 1000 WC using ordinary commercial brass. The only significantly thicker .38 brass I have is milspec picked up off a USAF range years ago. If you're wanting to load some spl high-pressure load in .38, the surest way to get brass is cut down .357 brass. But high-pressure .38s are safest to shoot in a .357 gun, so what's the point? The pressure in a revolver is taken by the cylinder, and a cartridge fired in the open results in tiny brass shrapnel.
I doubt that wadcutter brass would be thicker. As mentioned before military brass was thicker but when loading wadcutter projectiles in military brass, the bullet, being longer, would bulge the brass. Another issue with thicker brass, would be increased pressure (less power capacity). I have heard that +p brass suffers a similar fate.
Brass originally loaded with WC normally have 1 or two canalures on the case. I load mostly Wad cutters for my model 52. Use Nickle RP brass as it is easier to find.
Military 38 Spl. brass is thicker, have never measured case thickness on other 38 Spl. brass. I load for Colt Mid Range that has a " threaded"chamber to delay ejection to assist in operation. One would need an wall thickness mic.
38 Special wadcutter brass typically has two cannelures rolled into the case and the thinner case mouth section of the brass extends deeper into the case to allow for the longer hollow base wadcutter bullet to seat without significant deformation. At least that is my understanding of the significance of the special wadcutter brass.
I just sold a case of once fired Winchester nickel for $100 shipped and had to drop the price. They filled a medium rate box to the top without the boxes. I use the smooth brass for wad cutters. I tried to measure but they were the same thickness. When new they may measure a different taper. Remington hbwcs measure different on each end.
Many years ago I still had factory wc ammo, that is still holding on.
Here is a picture of two factory wc cases with a reload of mine on the right.
If you are a old hand at loading, you might remember the bullet company
that is in the center of these loads.
They are now gone and I finally ran out of their lead bullets.
I posted this picture so you could see where the crimp stamps were located on the cases.
I have about 10 or so boxes of .38 Special "Super Match Mid Range" from both Winchester and Remington that was made in the 60's. It is for a rainy day. I have thousands of U.S. military brass cases from the 50's - 90's. Also a thousand or so double cannulured nickled cases from the same time frames. All seem to shoot my reloads with no problem either in my J, K, or N frames and my 52-1 or the Colt Mid Range that I sold to a board member here. I juse use a 148 grain DEWC for the revolvers and a HBWC 148 grain for the autos. I use 2.8 grains of Bullseye with a standard pistol primer. All will go in one hole if I do my part. I keep reloading the cases until I see cracks, then throw them away.
I thought, IIRC, that wadcutter brass was thinner and had two cannelures. But normally I would use what ever I picked up for my 38 Special reloads. When I first started reloading I sorted all my brass by headstamp, but for most of my calibers reloaded it proved to be not necessary...
Winchester and Remington had the two cannelures, Federal has a single one about 1/3 of the way down. The Hornaday I saw was head-stamped 38 Special w.c. Any others might be all over the map.