The possibility of connection to JAWS and NVDA through Java Access Bridge is needed. As far as I understand there is no support for Assistive Technologies in JavaFX (unlike in Swing).
I found the project called fxaccessible, but it is not well suited in my case.
How can I enable support for assistive technologies in a JavaFX program?
At least some of the controls have source code for accessibility related implementations.
The great majority of JavaFX code is open source, so you can download the source repository and check the extent of accessibility integration in the current builds and also contribute any modifications you make for accessibility support back to the JavaFX project if you desire.
The JavaFX roadmap states:
Oracle is committed to providing accessibility support in its product offerings, and JavaFX is planned to be fully accessibility compliant as part of the JDK 8 release.
So one course of action is to wait until JDK 8.
That is rather risky though as the roadmap also includes a disclaimer:
It is not a commitment to deliver any material, code, or functionality, and should not be relied upon in making purchasing decisions.
I'd advise posting your question to the openjfx-dev mailing lists where the Oracle developers and project management can advise on the current status (and possibly rough future timeline) of the accessibility integration into JavaFX as well as the current technologies and strategies used for the accessibility implementation in the code base.
Also check the javafx issue tracker for issues related to accessibility.
The fxaccessible project you reference in your question (as well as being not well suited to your case) is not really applicable for anybody anymore as it is based on the obsolete JavaFX 1.x scripting technology and there is no way it can function with modern JavaFX 2.x systems which don't include the ability to run JavaFX 1.x script.