Its cool to check out any SR tailor, even if you're not looking to place a bespoke order. As Manton said, they're very retail-oriented and used to people rolling in just to buy a tie or square. A+S is very good about giving a full tour even if you're just a walk-in, and the people at Poole, W&J, and Dege are very friendly. Gieves' store is just beautiful. And don't forget all the adjoining streets that house the smaller firms - I would recommend stopping by both Meyer & Mortimer (Sackville) and Hayward (Mount).
Be sure to stop by the Arcades - Cleverley, Santa Maria Novella, Benson & Clegg etc. are all there. And Budd should be a definite stop - really is a fantastic little store. On the more casual front, I would suggest checking out Albam.
P.S. - An English tailor recently told me that Cifonelli is opening a London outpost on Sackville. Anyone know about this?