You know, guys, I suppose it may be entirely possible that the regulator circuit could be designed in such a way as to not be particular about the tube it uses, and I'll be the first to admit that I don't know the specific circuit well.
But, there are some pretty large differences between a 12AU7 and the AX7 and AT7, as I'm sure you know, and it is difficult for me to conceive of either of the other two being a drop-in replacement for the AU7 in a regulator circuit. I would have to think that the operating points would need to be too different.
Having said that, I would be pleased to learn that I'm incorrect, because the explanation would undoubtedly be very instructive.