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.
