We aim to keep this thread updated with the latest news, and will add or update any issue as we can.

Battle.net accounts unable to access PTR ("Login server busy"):
Latest news from our investigations is that this issue is resolved for the majority of accounts affected. We are aware that the issue is still unresolved for some affected accounts, and most likely this issue will unfortunately remain for the rest of the Public Test phase for patch 3.2. To help prevent this issue recurring for future patch testing, please be sure to check out this message and FAQ:
Battle.net Login Help

Character Copy queue speed
On Friday 26th of June we signifcantly increased the speed at which both premades and live characters are copied.

Cannot access the PTR (Password / login details issue)
If you have created, upgraded, changed passwords or converted your account recently it is likely you may be unable to enter the patch 3.2 public test, and may need to wait for a future time to participate in our PTRs. This can be quite normal for PTR testing phases. We do, however, appreciate your interest in helping us test upcoming content, and hope you will try again for a future patch.