Actually, it's Marks and Spencer, not the other way around.
I second the recommendation for Harrod's. You also can't miss Fortnum & Mason. Fantastic tea
and some specialty foods.
Go to Liberty of London if it's still there (they filed for bankruptcy, and their NYC store closed some time ago, I'm not sure about the London store). They have several nice fabric patterns that they've developed (mostly paisleys), and you can get skirts, scarves, ties, etc. in those patterns. They also have an entire section devoted to cross stitch and needlepoint kits (a hobby of mine).