It is in theory possible to do the PPL in the uk in three weeks - I managed it in 4 winter weeks and that was not flying at weekends.
I'm not up to speed on the cost differences between here and there, but I doubt that there will be much in it (when you include flights, accomodation, check rides when you get home etc). Learning locally will get you familiar with local procedures, landmarks and radio work. That will make a big difference to your first few post ppl trips.
Also if you get to the end of your three weeks and you are not ready for the test, how would you complete the course?
And if you are going for a PPL and a holiday - three weeks is not nearly long enough.