Thanks - some useful thoughts there.
She and her husband share the ISP. He does a lot of work from home; she is out and about a fair bit. Hence he wants his mail to come to the home PC, she wants also to be able to receive hers on the iPad. Both can't connect at once - which is where I came in.
green granite - The iPad doesn't remove mail from the server - it leaves it till the desktop machine pulls it off, so there's ultimately always a master copy on the home machine.
Milo - I tried Wanadoo on one and Orange on the other. Same behaviour.
Gertrude - I thought that too, but after several hours trying all sorts of ideas to find out why things weren't working, I can confirm that if either machine is in "mail" mode, the other one can't connect. Close down OE on the desktop and the iPad can get mail; otherwise it keeps asking for the password and saying it's wrong. Likewise the other way round. That isn't for a few seconds: it's for several hours - I know, I've done it.
Saab - that's how every ISP I've ever used does it. Except Orange, apparently.
Mike - she doesn't have her own domain yet: that's a bit further up the learning curve, although I think she and husband will be going there. I offered her (indeed, set up) POP3 addresses on one of my domains, but that had the problem of not being "her" address. She also has a Gmail account (IMAP) but doesn't like that at all. IMAP isn't good when travelling around out here in the sticks where 3G connections come and go and WiFi is sparse.
It sounds as if nobody here uses Orange, or has tried to connect multiple machines. Meanwhile, an Orange person has contacted me and is looking to find an answer from the techies.
If that reveals no solution, I suspect a change of ISP will be the outcome.