However, old maps suggest a slightly different story.  
An Ordnance Survey Town Plan of 1889 https://maps.nls.uk/view/231287205 suggests "Pope's Head Alley" for the eastern one and "Peter Lane Passage" for the western.
An earlier OS Town Plan from 1851 https://maps.nls.uk/view/231287106 also has "Pope's Head Alley" for the eastern one but no name for the western.
It's possible that the OS got confused in the 19th century, but it's also possible that strict modern naming rules weren't adhered to.