Many thanks all. We have, in the past 15 minutes, managed to get the darn thing working (after three and a half days of some serious head scratching and highly colourful language).
The box that is hosting the servers uses an HP NC375i Integrated Quad Port multifunction gigabit card, and, for one reason or another, two of those ports were plugged into our switch. The management network was going out through one port, and the pool of servers was going out through the other. We've unplugged VMNIC1 and forced everything through VMNIC0...hey presto! No idea why this works (especially as when we just moved everything onto a single nic and left the other port plugged in there was no issue). It's not ideal, but at least it makes it usable now.