I am a US citizen but reside in the UK. I'd like to take advantage of a tax exempt "Qualified birth and adoption distribution" from my rollover IRA. Do I qualify?

You will still be eligible to take a distribution of up to $5,000 penalty-free, (but not tax free) even though your child is foreign born.  You can also claim expenses paid for the adoption for the adoption tax credit, if the adoption is otherwise eligible and your income falls within the limits for the credit. 

Your IRA custodian may request a copy of paperwork to process your distribution, but most do not.  You will not attach that to the tax return if you are only using the penalty-free distribution.    It is also no longer required to submit proof with your tax return for the adoption credit, but you should keep information related to the adoption with your tax records for at least three years. 

The only difference with a foreign adoption for the purposes of the credit is that you can only claim the expenses once the adoption is final.  You can learn more about that credit HERE.