BSA stands for British Small Arms. They were one of Raleigh's main competitors until Raleigh absorbed them (like they did with almost every British bicycle manufacturer).
As far as I know, noone is making exact replicas of these bicycles, at least in North America, but plenty of the originals are still roadworthy. They can be found quite cheaply at garage sales and in classified ads. Your Phillips was one of Raleigh's many rebranded models, so you will see a wide variety of names on nearly identical bikes. The highest quality were sold under the Raleigh, Rudge and Humber brands, with Phillips being a step down. Take a look at Sheldon Brown's page on English three-speeds for the full details.
If you're set on a new bicycle, you could check out Flying Pidgeon. These are commuter bicycles imported from China, and are very similar to the Raleigh/Phillips/etc roadsters. Several manufacturers have also started coming out with modernized three-speeds in the last couple of years. Off the top of my head, the Schwinn Coffee (I think this has been renamed) and the Kona Africabike are probably the closest to the originals. The utility forum has a thread title "Bring Back the English Three-Speed", which mentions most of these newer version.