(see previous post!) has been solved, but I will confess to not understanding why the solution works.

The router is supposed to have a default MTU setting of 1500. It was set to 1452 instead. Restoring it to 1500 has made it work again – the PCs will now work with Hotmail et al.

But why?

I suspect, though I haven't actually checked, that the PCs always had an MTU of 1500. All having a lower setting on the router should have done is mean that some packets were split when they otherwise would not have been. But that's what routers are for – to split things up and stick them back together again…