As far as I know the first all steel body was the 1912 Hupmobile. The bodies were engineered and built by Edward G Budd, Joseph Ledwinka and their team at Hale & Kilburn in Philadelphia. The Hupp manager Emil Nelson came to Hale & Kilburn in 1909 because no shop in Detroit would take on the job. When Hupp got new management they went back to composite bodies for the 1913 models.
According to Coachbuilt the first major contract Budd got for his all steel bodies was an order for 2000 bodies for Oakland. I don’t know that led to follow up orders from Oakland. John Willys ordered 2500 bodies for Garfield, which was building trucks. Willys thought a passenger car would be good addition. It wasn’t. Garfield went bankrupt and I don’t know how many, if any, actually were sold. The Dodge Brothers were the only ones that went all in (almost) for all steel bodies.
The development of welding and paints are of course important, but the focus of this article was on the new thinking and engineering that went into the Dodge Victory Six. It is possible that Wolseley in England were using this technology before Dodge did, but only about 500 Wolseleys if this type were built over a period of six years, which was far from the numbers necessary to make all steel sedans profitable. The Wolseley bodies were made by the English subsidiary of Budd’s, Pressed Steel. A fun fact is that a modified version of the Wolseley body was used for the Ruxton.