I've done this in my 156 SW about a year and a half ago.
Everything is just bolt on, you just have to change the ecu, code box and key chip.
As the 1.6 doesn't have the variable intake manifold, you need to add the plug to the loom.
You just have to remove the plug from a used old loom and then insert one wire in the ecu plug and solder the other wire to the common +12v of the other plugs, like the one in variator solenoid or the injector plugs.
There are some differences, though.
The top engine mount is different, you must buy one for the 2.0.
So is the twin cat section just bellow the exhaust maniflod. The one the 2.0 uses is taller, because the engine bottom end of the 2.0 is taller than the 1.6 one.
You can use the 1.6 twin cat (as I am currently using), but you won't be able to bolt its bracket to the engine block. This causes that sometimes the flexi section bellow the twin cats rubbs the oil sump.
Another thing that is different is the vacuum pipe that goes from the inlet manifold to the power brake, because its socket is differently placed in the 1.6 inlet manifold.
As for the gearbox, you can use the 1.6 gearbox, but you should use a 2.0 clutch.
With the 1.6 gearbox it is faster, but has lower top end. Gas mileage will also be worse.
I used it in two trackdays and in the straight, a 159 SW Ti and a 147 (both with around 180 hp) couldn't pass me. I couldn't get away from them, but my car was standard 150 hp. The gearbox was doing the rest.
Notice, that in this straight we could only get to 170 Km/h, I'm sure that from 190 Km/h they would leave me behind.
My 156 is a CF3, but it is the same in all versions.
I'm almost certain that in 147 the work will be the same.
Here are some pictures of when I added the plug to the variable intake manifold: