Did the U.S. Army wear a standardized PT uniform during and immediately after WWII that included "shorts" and "running shoes"?

Same here, while in the Army 72 to 75, no shorts just fatigues, T-shirt, black combat boots, and front leaning position was for anything out of the ordenar.
At Bragg, in walking formation to do the daily run, if we turned to the left, it was a 2 or 5 mile run on the pavement, to Smoke Bomb hill to tease the SF doing their PT.
If we turned to the right, it was the Airborne mile run into the wooded trail run, but it really wasn't a 1 mile run in the woods, it was a 2 mile obstacle run, up hills and down hills, and a sweat box run.
Oh, the "Daily Dozen" was alway first before the run, and the daily run to the mess hall for chow after being dismissed from PT, proved how fast you were, depending where you were in line at the mess hall.

SGT (Join to see) - I really don’t remember our runs being that involved only a lot of them,what I would have given for running shoes,our admittance to the mess hall required,questions on General Orders and Monkey bars,Monkey bars also on exiting then falling in for formation to run to barracks,Be well Brother

SGT Philip Roncari - Thank you, brother Philip

Shorts and t-shirts were used but extensively not in the manner as we know it today. Running shoes were not a thing. The service shoes AKA boondockers is what was used as the running shoes. You'd have a couple pairs, field pair and garrison. PT shoe was not used until late war and after because the army switched to the buckle boots during ww2 and later to the M1948 russet boot to replace the service shoes. Paratroopers were still issued the same gear as the regular but was usually stored for inspection purposes only.

Hope that helps.

There was no PT uniform when I started Army BCT in NOV69. When PT was conducted, we were wearing OG-107s (cotton OD fatigues) with the standard black combat boots. In summer, the shirt would be removed, neatly folded and placed on the ground, and whatever cover was being worn would be placed on the shirt. In the winter, the shirt was un bloused, but remained on the soldier.
This was for organized PT as well as the PT test.
LCDR Joshua Gillespie SGT Justin Anderson PFC Dagmar Youngberg SFC Kenneth Hunnell SGT (Join to see) SGT Philip Roncari

warrior

And man did those boots cause shin splints!!

we got two ea. sets of sateens, first angle pocket jungles, black boots, all-purpose uniform, white skivvies and t-shirts , they did have some shorts and pt shirts for organized runs, house mouse washed it all night, mostly boots and utes for all else, including PFT. We were lucky, the 60's PRT was a bear.