John Thompson Motor Pressings Ltd supplied the formed steel components for the MGA chassis (prior page). The chassis were then welded together by teams of two men at the MG factory at Abingdon (this page).
This video is from 1950, apparently welding together an MG TD chassis.
The father of Colin Grant, the deputy editor of the MG Car Club's magazine "Safety Fast", was one of the people assigned to welding the chassis into the shape we now recognize. A few years ago (prior to 2018) Colin wrote some of his recollections of his father's time at the MG factory in Safety Fast. If you want more information you can email him through the MG Car Club. -- colingrant@mgcc.co.uk