When an amended return is filed, how long does it take before that refund amount is released?

Do not expect quick results from amending. It can take four months or more for the IRS to process an amended return.  

You can watch for information here:

https://www.irs.gov/Filing/Individuals/Amended-Returns-(Form-1040-X)/Wheres-My-Amended-Return-1

It can take up to 16 weeks or longer for the IRS to process an amended tax return.

Go to this IRS website for the status of an amended return - https://www.irs.gov/filing/wheres-my-amended-return