I signed up for Google Apps for My Domains a second time (for another domain). This time the process was different I think. Instead of waiting for (presumably) human review, it asked me to validate I owned the domain by adding a CNAME.
That was the second option and was easiest for me to do. The first option was to put a particular web page on the domain web site.
I've pointed the domain's MX records there and I'm now waiting for them to update in DNS.
I had immediate access to the domain manager panels at Google and was able to set up the login page, accounts, etc.