I'm still having all sorts of problems getting online with this laptop (thank goodness AW isn't affected!) but I might have tracked down the cause.

A bit of googling revealed a lot of problems with this laptop model, and its network adaptor: it's an Asus K55A, and it has inside it a Qualcomm Atheros AR9485 wireless network adapter. That Atheros adapter proved problematical for a lot of users by the look of it, so I'm going to look for a better driver for it and see if that resolves the problems I've been having. Fingers crossed.