The only way of changing the capacity of an engine is to either change the bore of the cylinder or changing the stroke of the piston by changing the diameter of the crankshaft. This is a major exercise.
I've checked the specs on the One 1.4 engine and it has a bore of 77mm and a stroke of 75mm, whereas the 1.6 has a bore of 77mm and a stroke of 85.8mm. This means that the increased capacity is provided by a "longer throw" crankshaft. That's a
major job and not to be done lightly and that's before changing all of the ECU software to match.
Upgrading an R50 One to R50 Cooper engine spec is sooo easy, the R56 One to Cooper change is going to be a major challenge that simply can't be worth it
I guess BMW would (understandably) prefer One owners had no option but to return to the dealer to upgrade the entire car for more power, rather than (on the R50) talk to an aftermarket engine tuner and, for about £300, increase their output to 123bhp by a simple software upgrade.
Tigger.