Large Noses Run in Our Family...

Inspired by Booky Goatherd's excellect picture of Admiralty Arch, in London, I took a slight detour on my way to work this morning, to blip one of architecture's more mental idiosyncratic features.
Reader, as you travel from Buckingham Palace to Trafalgar Square, pause awhile as you pass through Admiralty Arch & look up to your right.
Why does the arch have a nose?
The theory is that it is a humourous tribute to the Duke of Wellington, who, apart from foiling the machinations of Napoleon & stout footwear, was well known for his large hooter. The nose is set at such a height that, by tradition, a rider on horseback can rub it for luck as they pass.
But as the arch was completed in 1912, & Wellington died in 1852, can that really be the full story? Any ideas?