Phoned up Durham this morning, to check on the status of my transcript. Got shuffled around a bit, but then spoke to the woman in charge, who said that my transcript was now ready, and she had it in front of her. So, she's going to post it to me today (I confirmed that she had the right address), and she also said that she'd fax me a copy, which will help, since I can then start writing my application while I'm waiting for the letter to arrive. This fax hasn't actually turned up yet, but I'll give her the benefit of the doubt - maybe there's a big queue in her office or something.
Anyway, it looks like things are finally moving, which is a relief.