The school district we were in didn't have any problems with their UK jabs. If I can think back accurately, they started school before they had seen the paediatrician, and that was fine - there is a two week grace period in Ohio. The school counsellor even recommended the paed we should go to.
The doctor reviewed the printouts that I got from the UK Practice and wrote them herself into the school form. They needed to have the TB skin test (heaf), since we were from out of state, but I don't think the negative result ever made its way to the school. It was basically forgotten.
If the doctor in the US thinks they are short of jabs, they will give them. For example, when we moved, it was standard in the UK to have four rounds of the triple vaccine + polio, whereas in the US, they did it in five. Therefore, the doctor gave them their fifth jab.
My two eldest kids didn't need the Hep B jabs, because the requirement was for all children entering KG that year, and the younger of the two was entering first grade. Now, 8 years on, the requirement is for anyone entering up to 8th grade.
My younger three had to have the Hep B, so I had all of them done at the same time, even though it was voluntary for the older ones.