No, the first 16 and 18 size Washington Watch Co. signed Illinois watches were produced in about 1902 (around 1.6 million serial number) and Illinois was no longer making keywind watches at this time.
The 18 size line of Washington Watch Co. watches were ...
"Lafayette" - 24 jewel
"Army & Navy" - originally 19 jewel, later 21 jewel
"Senate" - 17 jewel
"Liberty Bell" - originally 15 jewel, later (and most) 17 jewel
"Potomac" - 17 jewel
... and these were all nickel finish stemwind watches.
Now some early model 18 size Illinois movements were signed "Washington" or "Washington, P&K", but I've never seen any evidence these were made for Montgomery Ward.