Hi,
I'm assuming you'll be travelling light or leaving your bags at the aiport.
In that case i'd take the heathrow express into Paddington, then cab it to where ever you want to go, it will save you a LOT of time compared to the tube....
I would buy no clothes at all in england, a good rule of thumb is that whatever costs a pound here costs a dollar there, including Burberry -which no one in london would ever wear unless they like hanging around car park late at night....