I ran into the same problem when replacing my iphone and macbook at the same time and this drove me crazy. Google prompt seems to rely on the device that the app was previously used on as 2FA method. So if you had a Google app on your phone, only that phone would get the prompt. Fortunately, I was able to view my gmail emails via the iphone's native mail app. However, I was unable to log into my Google account from the browser or Googles apps (gmail, docs, drive, maps, etc..). I was also unable to log into my gmail via Chrome or Safari on my new laptop. So all I could do was read emails, but not change any settings or even view emails anywhere else besides my iphone.
I ended up fixing this by accidentally downloading a 3rd party app on my macbook (from app store) called Mia for Gmail. While entering my gmail login credentials in Mia, it opened up a chrome browser window and asked me to login in from there. Amazingly, it worked! Mind you, it didn't work before when I tried logging in via chrome. So there must be something about the 3rd party login process that bypasses the Google prompt requirement. Considering the iPhone's native mail app and the Macbook's Mia 3rd party email client both were able to access my gmail (and whole google account for the latter), I suspect this has something do with non-Google owned apps having a different authentication process.
I have no idea if this will help anyone, but I figured I post my exact solution to this problem. Google is a horrible company.