Can anyone tell me why the APR will be slightly different when first payment (auto loan) at closing is increased from 30 days until example 60 days?

The longer it takes to make the first payment, the longer that interest accrues on a higher initial principal balance resulting in a higher APR.

Thanks so much for your response:

Do these calculations make sense?

Rate – 5.390

APR

24 days until 1st payment – 5.386 - Total finance charge - 2,624.80

30 days until 1st payment – 5.385 - " " - 2,643.40

45 days until first payment – 5.389 - " " - 2,689.60

Why would the APR on 24 days be higher then 30?

You need a lot more information than that to run APRs. And why do you care unless your APR is off by more than .125%.

Just trying to understand why

Your APR is going to be based on the loan amount, the prepaid finance charges, the date on which the interest begins to accrue and the required payment streams, which will give you your units and odd day periods. It comes down to the math that is used. See Appendix J in Regulation Z. Without knowing those factors, we cannot even begin to see if the APRs you are quoting are correct.

This is an "apples and oranges" problem. The math in Appendix J uses the actuarial method to calculate the APR. That method assumes equal payment intervals, and compounding of long-odd-days' interest at the end of the odd period. Meanwhile, you are most likely using the "U. S. Rule" method to calculate your payment schedules. The "U. S. Rule" is a daily simple interest calculation method that reflects the exact number of days from payment to payment, and does NOT permit compounding. If you're doing both sets of calculations correctly, the APR will almost always be slightly different from the nominal interest rate.