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There's no better place to enjoy an Afternoon Tea than at Fortnum & Mason (...well, apart from The Ritz).
Fortnum's is the closest that a shop comes to being a 5-star hotel. They even hold a number of Royal warrants to supply Buckingham Palace with groceries -- that's how luxurious it is. So how about dressing up and having an Afternoon Tea in their world-renowned Tea Salon?
Fillings inside their finger sandwiches include: Suffolk Cured Ham with Mustard, Cucumber with Mint Cream Cheese, Cotswold Legbar Egg Mayonnaise, and Fortnum's Smoked Salmon with Orange Caper Butter. Or how about a fruit scone served with Clotted Cream and a choice of Lemon Curd or Strawberry Preserve? Or tea cakes and pastries served from the ornate Cake Carriage?
You can check out a typical menu at fortnumandmason.com (.pdf file).
Note: The Diamond Jubilee Tea Salon has a dress code. Guests are asked to dress "smartly". Shorts, flip-flops, wellington boots, sportswear or beachwear are not permitted.
After you've finished your Afternoon Tea there are plenty of other things to do nearby. Piccadilly is home to the Royal Academy of Arts and Burlington Arcade -- and Piccadilly Circus is just a short walk down the road. If you time it right, then you may even be able to squeeze in the Changing the Guard ceremony beforehand.
If you really want to treat someone, then how about booking an afternoon tea inside the Palm Court at the Ritz Hotel instead? It costs a bit more than the one at Fortnums, but it's easily the best one in London.