What is the process for the new, extended child tax credit for 2020 returns?

That CTC increase is not for 2020;  it is for 2021.   Your 2020 return does not have to be amended for it.  More information will be available soon; the details of how it will be implemented have not yet been clarified fully.  There may be monthly payments going out beginning in July 2021, with some of the CTC still left as a refundable credit on your 2021 tax return next year.

https://ttlc.intuit.com/community/tax-topics/help/how-will-the-stimulus-package-impact-me/01/1393859#U1393859

The increased Child Tax Credit is effective on the 2021 tax return to be file next year, not the 2020 tax return.

Any advances on the CTC will be handled by the IRS.

See this article from Kiplinger's for a detailed explanation - https://www.kiplinger.com/taxes/602431/child-tax-credit-2021-who-gets-3600-will-i-get-monthly-payments-and-other-faqs