My company was purchased by private equity who is choosing to pay all vested Options and RSUs in full in Sept. If taxes are not taken out, when would the taxes be due?

Quarterly estimates for income paid after Sept 1 are due January 15.

However, when you prepare your tax return, you may want to check the penalty calculation anyway.  The IRS assumes all income is spread out evenly over the year, so they expect taxes to also be paid evenly.  That means that for a $10,000 RSU awarded taxed at (let's assume) 22%, the IRS will want to see payments of $550 each paid on April 15, June 15, Sept 15 and January 15.  To avoid a penalty for not making the earlier payments, you can use the "Annualized income installment method" which divides the year into 4 quarters, allocates the income accordingly, and calculates the tax due each quarter.  By showing the IRS that your income was unevenly received, and you paid the right amount of tax for each quarter (via a combination of payments and withholding), you will not be assessed a penalty.  Turbotax includes this calculation under "special circumstances."

https://www.irs.gov/payments/underpayment-of-estimated-tax-by-individuals-penalty