Is it currently safe to travel to Bali in terms of terrorism alerts? Are there any official warnings not to go there?

You would need to check with your country's Dept of Foreign Affairs for official Alerts.

My cousin & his family are leaving Australia tomorrow for a month's stay. I'm a reasonably frequent visitor.

As far as I know there isn't any current escalation of the normal general warning about Indonesia & Bali.

i just left bali and there are no problems at all...

I was in Bali in March and there had not been any unrest nor any "noise" that would cause one to suspect a problem was brewing. You just need to avoid the airplanes in the Us for the underwear people or the two US New Jersey residents that were on there way for training to do us harm.
Go and enjoy. It is a beautiful part of Indonesia as is Central and East Java.

Was in Bali recently, no worries about safety when I was there.

My wife, my 2 year old daughter and I were there in March - in Seminyak and Ubud. Nothing terrible happened. I think it's impossible to say that nothing bad will will again, but then, isn't that true of almost anywhere these days?

All any of us can really say is that when we were there, it was like this ____. In our case, ____ could be filled in with calm and normal feeling.

We who live in Bali always scoff at the travel warnings because of course anything can happen anywhere. That said, you have to be careful no matter where you are. That reminds me of a time I met a young Brit couple on a plane and I told them I was flying to LA. They asked me if I was afraid to travel there because there are so many bombings and drive-by shootings. Well, yes that does happen but not often, and you know how to lessen the risks by being careful in your conduct.

And yes, I am here now and feel perfectly safe.