Fix asset processing tests being flaky from hot reloading.

[andriyDev]

Objective

• Fixes flaky asset processing tests introduced in Add tests for asset processing under hot reloading and across asset sources. #21673.
• Previously the hot-reloading processor tests would just check for the finished processor state. However it was possible for processing to not start quickly enough in multi_threaded, resulting in the test still seeing the state as finished.

Solution

• Use a lock to block reading the source directory until after processing has started. This way the asset processor can't even check if there's work to do until the guard is released, so we can't miss the processing state.

Testing

• Ran it locally with singlethreaded and multithreaded mode.

[kristoff3r]

This is really clever!