Mafoofan - welcome to London, hope you have a great time. Should have quietened down a bit, post-Olympics, by the time you get here
My recommended walking tour would be to start at the Regent Street end of Jermyn Street and walk down to St James', in particular taking in John Lobb Paris, Fosters (for the bespoke samples), Edward Green, T&A and Emma Willis (for a great selection of Drakes ties in 9cm width).
Take a left down St James' and you will pass John Lobb London and James Lock just before you reach St James' Palace. I would also recommend a quick visit to Berry Bros wine merchant. Then head back up St James', take a right on Piccadilly - and walk on to Piccadilly Arcade where you will find Budd and Santa Maria Novella. Then cross the street and take in Burlington Arcade
Burlington Arcade is becoming a little bit touristy - with Jimmy Choo for Men set to open - but it still has some wonderful watch shops, a great pen shop, as well as the Globetrotter shop. You'll be needing some more luggage...
Coming out of Burlington Arcade, take a right - past the A&F monstrosity - pop down Old Burlington Street for A&S, and then back down again to get to Savile Row. Towards the end of the Row take a left down Clifford Street, where you will find Drakes (as well as possibly the new A&S haberdashery), Kiton and Lodger/Lutwyche. At Drakes, I find that the antique cufflinks and accessories collection is always worth a browse. Found some great abalone and rose gold links there a while back.
Next I would suggest walking down Bond Street to get to the Royal Arcade and Cleverley. At this point you have done pretty much all of 'classical' sartorial London. A couple of potential additions would be:
+ Sackville Street (off Piccadilly) for Kent, Haste & Lachter, Meyer & Mortimer, and my own shirtmaker, Sean O'Flynn
+ Vigo Street (between Sackville Street and Savile Row) for swimwear at Orlebar Brown
+ James Smith umbrellas on New Oxford Street
+ Mount Street in Mayfair - not just for Rubinacci, but also for Haywards, where Richie Charlton (formerly of Kilgour) is cutting a great 1B coat. Also worth having lunch at Scott's
I wouldn't bother with H&H - the last time I was there it was very uninspiring
In terms of other places of interest
+ I would stop by Dukes Hotel for a martini. Either with the house Plymouth or with Berry Bros No 3 Gin. If you feel more adverturous try the Experimental Cocktail Club
+ Lunch at Scott's (above) or Wilton's on Jermyn Street (though it is sadly now closed on weekends), or even Franco's (also on Jermyn Street)
+ Have dinner at one of Hix's places - either the Brewer Street Soho restaurant, or at the Belgraves hotel. Alternatively, you could go for Italian at Polpo or Bocca di Lupo, or Locanda Locatelli (the best classical Italian in London)
Have plenty of other suggestions, but this would be a good start