No, it's single stage to orbit; it's a poorly written article. It goes to mach 5.4 with atmospheric air before switching to liquid oxygen. The article makes it sound like it only ever reaches mach 5.4.
It's the BBC, so I'd say it's an averagely written article in their context.
Noticeably, the point of the Skylon and SABRE is that it's possible to use a common engine core in airbreathing and pure rocket mode, thus potentially generating weight savings.
I'm sure someone sat down and did the calculations. I have this feeling that a lot of the times when you actually sit down and work out the math for a range of assumptions, you generally come down to 'eh, maybe A, maybe B, so may as well try A'.