One thing I would say about this is: Only tell them what you absolutely have to, so don't tell them you are finishing up, or that you have a viva; just say that your PhD studentship is over and that you are available for full-time work, which your are. In my experience, and from what I seen on this forum, the JSA people tend not to know their arse from their elbow and if you get a 'Pauline' (the league of gentlemen) type, then any fragment of an excuse could be used to disqualify you, or at least disrupt your claim, and they are nigh on impossible to dispute with in any rational way.
Good luck