First Direct
The FD service, unless there's an option I've missed, doesn't send an immediate alert, but you can set what you want to know about, and when, then alerts go out around 7am the next day.
You can also set the thresholds for notifications, such as any transaction above 50 quid, every transaction if your balance is below £xx, or a mini-statement on a particular day of the week. So it's not quite instant, but definitely useful for knowing when something's happened in a reasonable time frame.