Some more observations on the flaky behaviour of email (SMS seems to always work):
Switching iDrive (ignition) off and back on again stops email working again (no icon, no emails listed under messages in BMW Office)
If you have iPhone notifications setup correctly, changing the Alert Style from Banners to Alerts or making any change to the Alert Style fixes the problem and you then see new emails as an envelope in iDrive and see new emails listed. You have to do this every time you get in the car as a workaround to the problem. Not sure whose fault it is but hopefully they'll fix it
When you first get in and switch on it takes about 1 minute to initialise before it can register new SMS or email. You see a "busy" icon when it's initialising
BMW X3 (F25) xDrive20d SE . . titanium silver . . titanium front grille . . 309s . . xline . . xenons . . black leather . . alu trim . . professional multimedia package . . dynamic package . . climate package . . electric seats . . folding mirrors . . servotronic steering . . DAB radio . . business speakers . . sun protection glass . . split rear seats . .