Doing all the flying in the UK would be incredibly expensive, and also very difficult due to the poor weather. I think going abroad would be one of the best bits?!
In theory, it should be more expensive doing all the training in the UK, but Oxford is still one of the most expensive training providers out there despite the fact that they do some of the training in the US. I know for a fact that the US based instructors are very well compensated for being a light aircraft instructor.
The year-round VFR weather in Arizona is definitely a plus though.