Sayoko will probably stay, but its dependent on how loyal she is to Diethard.
Lakshata is dependent on what the Black Knight's betrayal does to the Union of Nations. If the Knights betray Lelouch, and have the intent on using him as a bargaining chip with Britannia, then it will upset the Union, and by proxy, India. Lakshata would side with her home nation if it came down to choosing between the Union or the Knights, since she came to the Knights to provide technical support and gather data for her own ends.
If the betrayal of the Knights does not affect the Union in any way, and only negatively impacts Lelouch, then she'll stick with the Knights, because she won't care who she is led by.