First, remove what you can using some more Blu-Tack (just dab it on and lift it off again repeatedly). This seems to work best if the Blu-Tack you are using to remove the other Blu-Tack is warm and sticky. This can be done by rolling the Blu-Tack around in your hands for a while. This gets rid of a lot of the Blu-Tack, but there is still probably some traces of Blu-Tack sticking the carpet hairs together, or even worked right into the individual hairs.
The packet for Blu-Tack says that "On some surfaces Blu-Tack may leave an oily mark, but this can sometimes be removed by lighter fuel or dry cleaning fluid." I presumed that this might also help get remainders of Blu-Tack out of the carpet. I couldn't easily get hold of either of these (as I don't smoke and don't do my own dry cleaning), but eucalyptus oil seems to work nicely. Eucalyptus oil can be found in many sinus treatments (the ones where you are meant to sniff the fumes - the one I used was called "Olbas oil", and contained less than 25% eucalyptus oil). Wet the area with whatever you're using (you don't need to drown it, but I covered the whole area thoroughly). I then left it for a bit while I went downstairs to get a J-cloth, but I'm not sure whether leaving it helps or not. I then scrubbed with lots of water, and the remaining Blu-Tack just came off.
Warning - I have quite cheap carpet in my room, and I found that the oil also seemed to temporarily mildly affect the adhesive of the carpet. But using a well-soaked J-cloth seemed to just wash it away, and the carpet was back to normal once the water had evaporated. I simply recommend not trying to pick off any of the Blu-Tack while the eucalyptus oil is on it (although I can't see why this would be neccessary), as you can accidentally pick off hairs from the carpet. There is however absolutely no lasting damage, so don't freak out.
I realise that I have experienced a very specific situation, but I hope that it can give something useful to others in a similar predicament. Given that I'm not about to tread more Blu-Tack into my carpet in the name of research, I hope people submit slightly differing situations, and their experiences.