Swapping gearboxes

Hi!
I'm an Opel Corsa 1.4 16v owner, the car has a five speed gearbox with short gears, so short that i'm able to start in second gear, and that at 130 Km/h in the highways make the car to make 4,000 rpms, and it burns like eleven liters for 100 Km, so it's quite a lot.

I've ever thought that this gearbox is a limitation for this car, and I saw that Opel has built for his 1.4 engine in turbo versions a six speed manual that, looking for the gearboxes photos, are almost identical.
Now, I know that when you get your naturally aspirated car a turbo, you have to check that the gearbox is capable of manage the new engine torque, but in the opposite situation, would it be a problem for the general car performance?
In particular, i'm referring to swap the gearbox of the 1.4 100 hp turbo 200 nm car in my 1.4 90 hp naturally aspirated 130 nm (that will probably increase to 140/150 nm depending on the new engine map) engine.

The problem so is, would it bad affect the performance of the car? Are there bad consequences?
Thank you in advance.

Federico