The London Underground opened its 1st Tube station – Tower subway – August 2, 1870.   
No stamp of the first Tube station has been issued, so we’ll have to settle for a map of the modern London Underground.

London Underground Map
Part of the British Design Classics
Issued in 2009 by Royal Mail
Designer: Halpin Grey Vermeir

Now unused and mostly forgotten, Tower station connected the north and south banks with a 1/4 mile line. Excavated in 1869, the line opened officially August 2, 1870 and ran a few months before ending. The train (pulled by cables) could carry only 12 passengers at a time, and was failure from the start:

Sadly, the railway was prone to failure. It broke down twice during a press-only experimental run, which can’t have helped the publicity. Then a man named Thomas Francis Jannang was killed in an appalling lift accident at the southern end — his head crushed between the top of the lift and the edge of the ground floor. The company went bust within months. 
The Tower Subway: London’s Forgotten ‘First Tube Line’ | Londonist

The tunnel remained open for brave fee paying pedestrians, who periodically used it about 30 years, until it was closed. The tunnel was dank, dark, rat and robber riddled and prone to periodic leaks from the river above. The path was closed sometime in the late 1890s when it was superseded by the opening of the Tower Bridge station. 

The tunnel is still down there and since been re-purposed as a utility tunnel.