I had similar issues, I repaired EDI and never talked to NASH, so it did not trigger the EDI my love quest. At some point, talking to Mr. House triggered it. I then went to the Old lady at Gibson and talked about Helios one, waited 9+ days and never got a recording. However at Helios one, and talking to Rivas triggered the first two messages back to back.
The wiki does have some info on the quest,
http://http://fallout.wikia.com/wiki/ED-E_My_Love
It seems that there are many ways to trigger the messages. For me, the head scientist (Hilbert?) at camp McCarran triggered the first one, then the doctor at jacobstown triggered the second.
From there you just have to wait it seems, and you will eventually receive a message. Just one word of warning however, if you let them take ED-E for upgrading they will steal all of your items. I lost a dozen unique weapons that way, so be careful.