There are plenty of places to satisfy your sweet tooth in London. Here are some of my favorites, from funky ice cream to fine chocolates:

Paul A. Young Fine Chocolates | 143 Wardour Street
Chin Chin Labs (Ice cream) | 49-50 Camden Lock Place
Bubblewrap (Waffle desserts) | 24 Wardour Street
Ottolenghi (Dessert counter) | 50 Artillery Lane
La Gelatiera (Gelato) | 27 New Row
Monmouth Coffee (Baked treats) | 27 Monmouth Street
Bread Ahead Bakery (Amazing donuts) | Borough Market
Aux Merveilleux de Fred | 88 Old Brompton Road
The Hummingbird Bakery (Lovely cupcakes) | 133 Portobello Road
Pierre Herme London (Macarons) | 38 Monmouth Street
Tsujiri Matcha House (Matcha ice cream creations) | 47 Rupert Street
BobaJam (Cafe) | 76 Stoke Newington High Street
Paul Rhodes Bakery (Baked treats) | 37 King William Walk, Greenwich

You can also find plenty of sweet treat stalls at these markets:
Borough Market (Everyday, except Sundays) | 8 Southwark Street
Columbia Road Flower Market (Sundays) | Columbia Road
Brick Lane Sunday Market (Sundays) | Brick Lane
Pop Brixton (Everyday) | 49 Brixton Station Road
Broadway Market (Saturdays) | Broadway
Spitalfields Market (Everyday) | near Liverpool Street
Portobello Market (Saturdays) | Notting Hill
Camden Market (Everyday) | Camden Lock Place

And good chances are your local pub will have some amazing sticky toffee pudding!