This is all my fault for not doing my research before calling BT, although one would think that they should know better than anybody else about what's possible and what is not. Of course they don't, because after a brief search I found a BT forum in which they (as in customers, not BT) suggested to enable "port clamping" in the BT hub's settings. Sure enough, there is a menu called VPN, where it is suggested to enable this setting if you are having problems with VPN, which I had. Oh magic... now it works with no problem whatsoever... well, that is so far, because apparently it is unsupported according to their own sales department.
Lesson 1: Always do your research before talking to anybody
Lesson 2: Do not think that customer support knows what they are talking about
Lesson 3: For God's sake, do not believe sales people!
I'm tempted to send a letter of complain telling them that they should have known about the BT Hub option and that it almost seems they were just trying to make more money out of me... but I think I will pass just in case they decide to cut my VPN and say "We told you it wasn't supported". Shame on you, BT...