William 4th in 1851 census; King William IV in 1843 and just named the King William in 1854. This is earlier the Black Horse, certainly in 1832.
King William was King from June 26th 1830 until his death in 1837; I guess this was why this was previously the Black Horse!
Comparing the 1832 and 1842 street directories answers some questions, and also raises more questions?
Shadwell index
A listing of historical public houses, Taverns, Inns, Beer Houses and Hotels in Shadwell, London - in East London.
Residents at this address.
1805/James Paxton, the Black Horse, New Gravel lane/../../Holdens Directory
Bell's Weekly Messenger 29 December 1811 - case linked to the Pear Tree
Summary - Shadwell Police Office, Monday, December 23rd
John Harris stated on the night of the murder, Williams came to bed at a little past two. He found the muddy stockings and shoes which Williams afterwards took, and washed them. Williams once observed that Mr Lee's house, the Black Horse, in Gravel lane, was a convenient one for robbery, the outlet behind was so good.
Mr Lee said that he once caught Williams opening his till.
1829/S Taverner, Black Horse, New Gravel lane, Shadwell/../../Robsons Directory
1832/Taverner, Black Horse, 102 New Gravel lane, Shadwell/../../Robsons Directory
1833-34/Joseph Taverner, William IV, 100 New Gravel lane/../../Pigots Directory
1836/Stephen Taverner, William IV, 100 New Gravel lane/../../Pigots Directory
1839/Stephen Taverner, King William, New Gravel lane/../../Pigots Directory
1842/Stephen Taverner, King William IV, New Gravel lane/../../Robsons Directory
1843/Stephen Taverner/../../../Kellys Directory
1848/Stephen Taverner/../../../Post Office Directory
1851/Richard Morris/../../../Kellys Directory
1851/Richard Morris/Victualler/44/St Georges East, Middlesex/Census
1851/Susan Morris/Wife/36/St Georges East, Middlesex/Census
1851/Richard Morris/Son/14/St Georges East, Middlesex/Census
1851/John Morris/Son/11/St Georges East, Middlesex/Census
1851/James Morris/Son/8/St Georges East, Middlesex/Census
1851/Elizabeth Morris/Daughter/5/St Georges East, Middlesex/Census
1851/Ann Fox/House Servant/17/Shadwell, Middlesex/Census
1851/Ellen Lawson/House Servant/47/St Georges East, Middlesex/Census
1851/Captain Aberg/Visitor, Master Mariner/30/Sweden/Census
1851/Captain Lundstrom/Visitor, Master Mariner/33/Sweden/Census
1854/William Weston/../../../Post Office Directory