I have no personal knowledge of the British Small Arms rifles (BSA) but I do the Remingtons. This is more than the original question asked but it seems relivant to the way the thread has moved.
Mike Walker designed the Rem Mod 700 predicessors, the Mod.s 721 and 722, before the end of WWII to give Remington something to concentrate on after the end of the 1941-45 war. I think that was well before BSA used the enclosed extractor and "three rings of steel" designs but maybe not.
The Remington extractor is a stamped spring-steel ring that is carefully punched out, deburred and shaped to fit into the bolt face ring. It is not cast.
The rifle's design was intended to allow for less expensive production methods on new machinery that had been develped during the war, including such things as automatic screw machine lathes for cutting tubular steel receivers. He conceived the ideas for the seperate recoil lug, a round bottom action and bedding, the trigger mechanism, the first practical working methods for making button rifling barrels (and later helped get the Hart's started in making quality barrels), all his original designs, for those rifles. In the early 60s, Mike developed the 700BDL and ADL series, including the much nicer stock, for a better looking rig. Sales records were soon set. He did not approve the change to pressed checkering, top management did that to cut costs. (It always takes high level management to screw up a good engineering design, right?)
I believe it was Walker's co-worker, Wayne Leek, who developed the Mod. 600, 660, 788, etc, in the mid to late 60s. Leek included significant portions of the 700s better design/manufactoring features, including the extractor and round receiver.
Walker was, and remains today (even tho he's getting pretty frail) an avid bench shooter. In the 50s he modified the 700 into the Mod. 40X series of target rifles, at first for the US Army Marksmanship program in Ft. Benning, GA, and, later, the public. He develped the .222 for bench competition and it won most matches prior to the introduction of the 6ppc cartridge. Later, he gave us the factory versions of the 222Mag, the 22-250 and the 7mm Rem Mag. We riflemen and hunters owe him, and Remington in spite of themselves, a lot.