Why is my gearbox faulting?

I have a strange, intermittent issue, where my car (auto) fails to shift gears within the first 5 minutes of driving, but only during specific conditions. When it fails, there is a lurch forward, and a Gearbox fault digital warning appears. The SERVICE lamp also turns on. All I have to do though is pull over, and restart the engine. None of the warnings come back on, and the car doesn't experience a fault after that.

The problem: My car's gearbox faults

The conditions:

• Cold, or just warmed up engine.
• Going up hill

A mechanic has suggested I need a new gearbox - but I'm not sure it's specific to the gearbox itself. Before I spend a bunch of money on a new gearbox, does anyone have any ideas about what this is or how to fix it?