Well, I never used the Scene Builder nor SWT, but one thing I'm sure:
The JavaFX application thread is responsible for taking care of any existing JavaFX element. You can find details on how to properly handle the JavaFX elements here. What I think you have to do is make sure to access JavaFX elements using the JavaFX application thread, which you can do using the method runLater from Platform class. Take a look:
http://docs.oracle.com/javafx/2/api/javafx/application/Platform.html#runLater%28java.lang.Runnable%29
As always, we know we should start JavaFX applications from the main thread of a Java program. What you can do is to use the main thread in any point of your SWT application to call the initiator of your main JavaFX class. For example, at one point of your SWT application, all you have to do is call launch.
Do not forget, however, that the main thread (the one that should call launch) will be locked until your JavaFX application is finalized.
Hope this helps, and good luck. :)