Hopefully a slightly cleaner mounting now. Thanks for pointing that out.
As for the rest, there was a faulty solder on D5 (the wire from the circuit board wasn't properly soldered), and I've fixed it, so C is now glowing. Still no dice with A and B, though. Will continue to explore.
Brief update: I have 6.2V between 22 and 24, but 0 between A4 and A5 and between B4 and B5, despite the connection being seemingly OK (resistance is nominal between A4/B4 and the positive pad on the solder board, or anywhere in between, and same with A5/B5 and the negative solder pad. I'm wondering if this might be because I'm not fully plugging the tubes (am I right in assuming that, without the tubes, there is no circuit?), but I'm unsure of whether I'm doing things correctly especially because of the bayonet base which hides the pins, and I'm afraid to apply too much pressure, even if I know that the sockets can be a little rough at first). Anyhow, will continue to investigate, but any bit of advice would be very much appreciated