Yeah today in the office (where there is Wi-Fi but only for work machines - I use Tasker to detect it and silence my phone in the office only) it is definitely not getting Gmail in real time.

I wake it up and a 5-10 seconds later I get an email or two or more (or I force a Gmail refresh). At home as far as I can tell it is instant i.e. my Outlook on my laptop and my phone alert basically happen simultaneously. The facebook alerts are pretty rare. Maybe I'll temporarily put an alert on one of my more prolific friends on facebook to test.

Without signing up to some other social networking site what else can I try? Maybe installing an app from Google Play?

It doesn't appear to happen instantly i.e. put phone to standby by pressing the power button and send an email a few seconds later. It vibrates and Lightflow flashes to signify an email as expected. Not sure at which point it stops but I'll take a guess and say somewhere around 5 minutes.