Here are my favorite drinks - both with and without alcohol - in London...
I adore a good
Flat White - an espresso drink that's similar to a latte, but tastes sweeter and has less milk/more microfoam - alas, they're all but impossible to find in NYC. Lucky for me they were all over London.
The best Flat White I had all trip was from
Monmouth Coffee in Borough Market:

It's a rustic space with fast service and strong, short coffees. I highly recommend you work this cafe into your visit.
Awww
Monmouth Coffee Flat White, I heart you too:

The second best Flat White I had was from
Salt in Covent Garden. This is a touristy area, but independent this shop is a welcome break from all the chains in the neighborhood.
I told the friendly barista I was on a mission to find the finest Flat White in town and he said he'd knock my socks off. Salt's Flat White - avec swan(!):

Color me impressed by that serious latte art. I've seen a heart, and sometimes an ellipsis, but never a swan made out of milk. It tasted awesome too.
Honorable mention goes to
The Espresso Room in Bloomsbury.
It was chilly and a bit damp in London, so when nighttime rolled around I was in the mood for a martini.
My favorite martinis were from
Dukes Bar.
Dukes Hotel has a small, discreet cocktail bar off its lobby where Alessandro, their charming Italian bartender, makes some of the best martinis in the world. Alessandro has a small wooden trolley that he wheels out after you order, and as he prepares your martini tableside he tells you the story behind the drink, his inspiration for the recipe, or a fun tidbit about its ingredients. It's an old-school style of service I happen to love, and his drinks are KILLER.
I couldn't pick my favorite of Alessandro's martinis, so here are the two that tied for first place --
1. Ian Fleming's Classic Vesper:

No.3 gin, Potocki vodka, Angostura bitters, and Dukes amber vermouth -- I felt like James Bond, just for a second when I drank it.
2. Dukes Gin Martini:

Plymouth gin, Dukes extra dry vermouth, lemon oil -- classic and flawless.
My second favorite martini was at the
Lanesborough's Library Bar. This bar is a *must-see*. I sunk into one of the Library Bar's plush velvet sofas set amongst the volumes of leather-bound classics in front of a roaring fireplace. It's a quintessentially English and perfectly cozy space.
I had their signature drink, The Librarian:

Cognac, Visciolata Cherry Wine, Antica Formula Vermouth, and Drops of Peychaud Bitter.
Seriously strong stuff, but also seriously delicious... and served in a cut crystal glass with linen cocktail napkins. The ceremony, service, and ambiance made me feel like ordering a shelf full of Librarians. (Don't worry, I stopped at one.)
Honorable mention goes to
Nightjar's English Mule cocktail.
So whether you're into coffees or martinis, I hope you'll find this list helpful on your next trip to London.
Cheers!