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The GE Essential Characteristics Handbook and Smith Tube substitution book say 12 AU 7 can be replaced with a 12 AX 7. (NOT 12 AT 7 with 12 AU 7)
I have never tried that experiment . Before plugging anything else in, I would ask the manufacturer.
For what it's worth, your better off that your unit was designed for 12 AT7. There is less pressure on NOS versions than for 12 AX7. Prices for superb 12 AX7 is getting ridiculous.
Though each of the tubes you list are dual triodes with identical pinouts, they all differ from one another SIGNIFICANTLY in gain structures. Therefore, the use of any of these tubes in any placement in a high end piece of equipment should maintain use of that particular tube for that particular application.
As I remember in terms of overall gain:
12AT7 < 12AU7 < 12AX7
And often, 12AT7 are used in output driver/inverter (for class AB push pull designs) applications ahead of the power tubes whereas 12AX7 are usually used in gain stages.