(or Classic English Typefaces to be a pedant, but I'll stick with the more widely used word 'font' and sidestep the pedant label)...
There are several fonts that are very closely associated with London, England or Great Britain to varying degrees. This seems quite unique - I can't think of any other cities or countries so closely associated with a font in popular culture, but we stand to be corrected! If we've missed any you can think of, please let us know - we'll update this article over time.
The post-war period (1945 onwards) saw a change of direction for English design, towards a more modern but still distinctly English style. As explained in this University of Brighton article: "This shift from the strictures of high modernist sans serif lettering to a more relaxed, celebratory and very English style had been heralded in the pages of the Architectural Review in the pre-war years. This turn to a jauntier and more decorative visual language was part of a wider move towards the appreciation of vernacular arts and the peculiarities of English culture, in both craft and industrial forms."
Street Name Signs
There's no standard font used for street name signs - it seems that different councils and boroughs throughout England have been able to choose their own sign design and fonts. However, that hasn't stopped some London designs becoming classics: City of Westminster, The Royal Borough of Kensington and Chelsea, and City of London are perhaps the best known designs.
In 1968 the City of Westminster commissioned Misha Black of the Design research Unit to re-desihn their street signs. The iconic result has rapidly become a famous and often copied design classic, combining absolute simplicity and legibility.
Pre 1958 Road Signs
The Government appointed Anderson Committee was setup in 1957 to review road signs across Britain, in response to the needs of the new Motorway network. Two graphic designers were commissioned to design the new system of signage: Jock Kinneir and his assistant (and later business partner) Margaret Calvert. The new signs were first used on the Preston bypass in 1958.
You can download this font here.
Post 1958 Road Signs - Transport
Transport (and it's associated font Motorway) were the brilliant design work of Jock Kinneir and his assistant (and later business partner) Margaret Calvert. They studied the needs of drivers on the new high-speed roads and Motorways, and they studied readability at different distances, speeds and lighting conditions. Notably they moved to a mixed upper and lower case sans serif font, which they called Transport.
You can download this font here.
This font, perhaps the most-recognisably English font here, was commissioned by the Underground Electric Railways Company of London from Edward Johnston in 1913. It was subtly revised in 1979 to be slightly heavier, and again in 2002 and 2008 - the last set of changes undoing some of the previous revisions to certain numbers.
We chose Johnston for our logo and roundel as it's our favourite font (closely followed by Gill Sans), and we think it's probably the most English-looking font:
You can officially purchase this font here.
Gill Sans 
Gill Sans is often identified as the font used by London Underground rather than Johnston (see above). Seen apart it's an understandable confusion, but next to each other there are obviously significant differences.
Eric Gill designed Gill Sans in 1926, and was used by the London & North Eastern Railway (LNER) in 1929. Interestingly Gill had previously worked for Edward Johnston as an apprentice, where he contributed to the development of the Johnston font.
Gill Sans has seen an resurgence in it's popularity due to the widespread over-use and mis-use of the 'Keep Calm and Carry On' posters from wartime Britain:
Those of a nervous disposition should avoid reading about Eric Gill's personal life - to quote Wikipedia "his beliefs and practices were by no means orthodox." Oh well, he was an extremely talented type designer and artist.
Festival Titling 
Festival Titling was produced for the 1951 Festival of Britain in London. It's a strange font that isn't easily categorised, and isn't recognisably modernist - it's almost a reaction against the unhappiness and austerity of the war years, and is all the better for that:
"Designed by Philip Boydell, it was actually a sans serif font, but used tall, thin letters and was only available in capitals. It had a vague resemblance to bunting and benefited from an almost three-dimensional, leaning effect which created a subtle sense of movement when it was used.
All in all the Festival lettering was considered very successful and was an intrinsic aspect of the festivals popularity."
The full article is here: http://wharferj.wordpress.com/2011/06/08/typography-at-the-festival-of-britain/
Here's Festival Titling in action:
There is a group of fonts that feature prominently in post-war English life on public building signage. This is a group of (generally italicised) slab serif fonts, the best example that I can find being Kaine.
London Olympics 2012 
OK, this last one isn't a classic yet, but it will surely be a future classic - not widely appreciated but it very powerfully represents the London 2012 Olympics, and with the games having been a success, maybe this font will grow to be loved.
In some ways there are parallels with Festival Titling: this font is difficult to classify, and is heavily stylised (perhaps too much - it's so closely linked to the Olympics that whatever it's used for will immediately be linked to the Olympics as well). This may well limit it's usage.