When Dr. Trevor Grant is interviewing Carter in 1947, he's smoking a modern filter cigarette. Cigarettes in America didn't start having filters until 1954.
The C-47 shown still has the D-Day invasion stripes (1944) painted along the side of the aircraft. Since this is 1947, they should have been removed by now.
The United States flag, located in the room in 1947 Camp Eureka where the dance was being held, is the current one that has been in use since 1960, and therefore is the incorrect version for 1947. The current flag has 50 stars (for 50 states) in the union (row pattern: 6-5-6-5-6-5-6; stars offset by row). The flag in the dance room should have shown the pattern for 48 states in the union, with 6 rows of 8 stars shown in a grid pattern. Although the flag was not fully unfurled, enough of the union was displayed to verify the incorrect star pattern for the time period. In addition, a desk flag in the medical barracks also displays the incorrect 50-star flag in an earlier scene.
Dr. Grant pulls out an old fashioned radio tube and calls it a transistor. It is obviously a radio tube based on its shape and the fact that it lights up when activated.
When Dr. Grant picks Carter up, Carter tells him that he looks "more officious" in his uniform. Since officious is a somewhat derogatory term that means giving assistance where it is neither wanted nor needed, the word he should have used was official. Unless he wanted to present himself as meddlesome, which is a synonym of officious.
The Army Cargo Truck which is wrecked in the chase is an M35 2½ ton 6X6 vehicle often referred to as a "Deuce and a Half". This vehicle did not enter production until 1949.
P-38s were retired from USAAF service immediately after the war ended. They are, like the C-47, in "invasion stripes" markings which were used from June 6th to September 1944. The markings would not be used in 1947.
The premise of the season, the crew going back in time and altering the present, contradicts the rules established in the Season 1 finale where Henry going back in time to save Kim lead to the collapsing of the two time lines. This season's time travel doesn't end in the collapsing of time lines.
Jo's Army uniform isn't configured correctly. she should have a "US" insignia on each of her upper lapels and her branch insignia on her lower ones. Jo is a West Point graduate and should know this. She would be arrested for being out of uniform if spotted by an MP.