This might be some important info for you.
https://www.samstagsales.com/whitworth_table.htm
A.
Ah yes..."Whitworth"...the bane of American owners of classic British vehicles. Unfortunately, many Whitworth sizes have no direct SAE or metric equivalent. For example, the 1/4 Whitworth (also known as 5/16 BS or "British Standard") is awkwardly between a 1/2" and 9/16" or a 13mm and 14mm such that even trying the 9/16" or 14mm is almost guaranteed to round out and ruin a 1/4 Whitworth fastener, and my old Norton is festooned with the goddamned things.
One problem with Whitworth tools is that you really pay a premium for them, when you can find them at all. They're such specialty items that you often end up paying Rolls Royce money for a Walmart tool that you might then end up using once every couple of years, if that. That's why nothing quite makes this old cheapskate's heart skip a beat so joyfully as when I stumble across some old Whitworth wrench or socket when pawing through a thrift store tool bin. That's my "happy place" while the wife scans those clothes racks like a Terminator (cue the "duh-dump dump duh-dump" beat).
This is just a hunch, but if your 1-9/16" (or 1.5625") socket is just a hair too small for that nut (say a tenth of an inch or so across the flats), then you could well be looking at a 1. Whitworth (or 1-1/8 BS), which measures 1.670", just a hair over 1-2/3", or 42.42 mm across the flats. Sadly, that's one of those awkward Whitworth sizes, but also kind of a likely one. But instead of paying close to 200 bucks for a proper 1. Whitworth socket, you might get away with a 43 mm metric one for about a tenth of the cost if you stuffed a piece of paper and maybe a dab of valve grinding paste into the socket to take up only about that 1/2 mm of slop. Then a 43 mm might grip just fine. You'll also likely need a little drive adapter, since 43 mm sockets are typically available only in those larger 3/4 or 1 inch drive sizes.
Here's a 43mm with 3/4" drive on Amazon for less than $20.
So, go ahead and measure that nut. I'm curious to know what you've got there.