As an Englishman ,I can state quite categorically there is no hard fast rule or process to the so called " English Brushed Finish "It is simply a term to describe the appearance as is " Coin Finish".
It will depend on the state of the action , the person doing it and the desired finish.
Webley & Scott did what was called an emery finish , which was simply cleaning off the colour hardening with fine grade emery cloth .
Others use soft wire brushes on a polishing spindle to remove the vestiges of colour and to even up the overall surface .
Some will polish the action to a shine
Actions have been bead blasted to give a dull overall grey finish , pickled in an acid solution to remove rust and staining before any brushing and/or a combination of both .
Personally I have used all of the above at some time or another dependent on what was desired or needed to give the best outcome or customer request .