Thanks Igor -- for the benefit of anyone finding this thread in the future:
I checked voltage on the PSU and all seemed fine. However, I got a replacement PSU "just in case" (a 5V 3A UK plug wall wart off ebay, instead of the EU plug one that came with the OPi; just had to solder a 4/1.7mm barrel connector onto the cable) and miraculously with the new PSU the board now comes up fine!
Lesson learned: PSU quality is make or break with these boards, even when using a decent barrel connector. And the symptoms of dodgy power supply can be pretty random.