The key thing is that if the emails/bank statements prove 100% that you purchased the ticket before travel, I don't see how Chiltern can charge you a penalty fare. You have paid the fare and purchased the correct ticket (which you can prove), and been allowed to travel by two agents of the railway who were authorised to do so. The problems with the App is not your fault! I would therefore send an e-mail or letter to Chiltern explaining this.
I am not as well versed in these matters specific to the railway as others on here are, so please don't take what I say as verbatim. See if they think along the same lines as me. However, I don't see how they could charge you due to something outside your control.
Also, as a wider issue, as the railway looks to use technology more and more - e-tickets, m-tickets, digital railcards etc. they need to come up with ways in dealing with faults. Otherwise they are going to start prosecuting innocent people who encounter these faults. I shouldn't have to buy another ticket if my digital railcard app has a fault with it and won't work! What if I don't have the means to?