Siemens managed to talk us through getting online with Simatic Manager.
On the 840D, there are two rows of LEDs. Of the left-hand row, just the top one was on, solid green (+5V, I think). Of the right-hand row, the top green one was on solid (PR), the next one was solid red (PS), the next one was flashing red (PF), the next one was on solid amber (PFO) and the bottom LED was also on solid amber (-). The 7-segment display was showing 6.
The Siemens' technician had me turn rotary switch S4 from position 0 to 3, then press the Reset button. This didn't seem to do anything, so then he had me turn S4 from 3 to 2, back to 3, then back to 2 before pressing the Reset button again. Again, it didn't seem to do anything, so then he had me turn rotary switch S3 from 0 to 1 before pressing the Reset button again. I'm not entirely sure that I've remembered the exact sequence but it was certainly something similar. We ended up with the two top LEDs on (5V & PR) and I was able to get online immediately.
We're still not out of the woods though, as he informed me that we need to find some archive files that should be stored on the machine's B&R industrial PC and carry out a restore procedure that reinstates the PLC & NC software. We're hoping to enlist the help of the machine manufacturer in this step.