Could we please straighten out things not belonging to each other?
It gives me headache to mix up all different part.
1: there is a contract with an ATO. Under EU regulations you have to have this for centrally managing all paperworks ... up to theory exam!
2: there is theory lessons. You may have this included in the ATO contract and get classroom training from them, or you may choose remote learning facilities separate.
3: there is a theory certificate the ATO, or the remote learning facility, issues when the student ist ready for exam
4: there is theory exam. After theory certificate is reached, the student, or the ATO when part of the contract, arranges theory exam
5: there is hours to fly under supervision of an ATO
6: there is a point in time the student decides to go for exam
7: there is practical exam
Typically a contract with ATO covers paperworks (1), maybe classroom training (2), certificate of finishing theory (3).
Theory exam (4) is a matter between student and local responsible government.
As operating a plane is quite costly, timing unpredictable and breaking off education not uncommon, part (5) is fly first, then pay for you flights on the basis of used resources (flight time per minute on the plane plus used time of the instructor). Yes, there you also have a contract with ATO, but not fixed, but variable (using pre-Paid as model for flying hours is stupid, sorry).
Part (6) is solely on the students discretion, even though flight schools try to press students.
Practical exam again is like (4) a matter without ATO, even though ATO tells different and/or try to get their fingers in there.
So, back to the question.
Yes, you can always change ATO during any stage of education. It is just an announcement to government that you did change ATO (most common done at the receiving ATO).
You may not get back money for the first stage up to theory exam, as up to theory exam is typically up front non-refundable payment.
At second stage, the only requirement for flying is "under the supervision of an ATO" - and that is any ATO. If you are on vacation, have all your paperworks with you and decide to take a lessons on your vacation spots airport and ATO, you are free to do so and they will sign it to your books. You do not need to do the flying hours at your ATO, ANY will do. There also is no obligation to "get signed off for solo" as I heard several times from students. Many ATO do that, mainly to be on the felt safe side, to celebrate some traditionals and to keep insurance quiet, but it is not a formal obligation. When you collected your hours you pack all you data and the theory exam to apply for practical exam. You can do this at that stage without an ATO contract, as you do not have to be enrolled at an ATO later than theory exam ;-). I know, quite trickay and most ATOs wont tell you and it may not be advisable to go that strict way, but it is possible and straightens out power balance between student and ATO.