You will have a really nice tractor when you are done. At one time I thought that I would like to own one, then two guys in our local club restored one each. J. wonderjet is correct the engine in a Cockshutt 20 (also in a 20 coop) is the same as is the transmission.
The Oliver Super 44 & 440 have the pivot off center on the front axle and have one long and one short axle housings. The early Cockshutt 20 tractors had the 124 engine and was changed in a short time to the 140. Most have been bored out to the 140 size. The Cockshutt branch did modify some 20 Deluxe models to the offset configuration.
Enjoy the Oliver when completed and be sure to post a photo.