It typically takes 1-3 months between submission and viva, but I have heard of as much as 12 months occasionally.
I think it's perfectly natural to say to your dept that you have not decided what to do after the PhD. It shouldn't make a difference.
I would have a polite chat with your dept's head, if your supervisor cannot help.
The timing usually depends on having all examiners, chair and sup available at the same time, which is quite difficult !