It seems important in this context to distinguish between consistency of different systems
Overall, my consistency estimate for PA if anything has gone up in this context. Because if Edward Nelson tries really hard and fails to find a contradiction that's more evidence that there isn't one. How much should it go up by? I'm not sure.
On the other hand, this may very well make my estimate for consistency of ZFC go down. That's because although in this specific case Nelson's approach didn't work, it opens up new avenues into demonstrating inconsistency, and part of me could see this sort of thing being more successfully applied to ZFC (although that also seems unlikely). (Also note that there have been attempts by a much larger set of mathematicians in the last few years to find an inconsistency in ZF and that has essentially failed.)
One thing that this has also done is made me very aware in a visceral fashion that number theory and logic are not the same things even when one is talking about subsystems of PA. I already knew that, but maybe had not fully emotionally processed it as much as I should have as when I saw Nelson's outline, and then the subsequent discussions and had a lot of trouble following the details. The main impact of this is to suggest that when making logic related claims (especially consistency and independence/undecidability issues) I should probably not rate my own expertise as highly as I do. As a working mathematician, I almost certainly have more relevant expertise than a random individual, but I've probably been overestimating how much my expertise matters. In both the cases of ZF/ZFC and the case of PA, reducing my confidence means increasing the chance that they are inconsistent. But, I'm not at all sure by how much this should matter. So maybe this should leave everything alone?
So overall I'd say around .995 consistency for PA and .99 consistency for ZF. Not much change from the old values.
We've discussed Edward Nelson's beliefs and work before. Now, he claims to have a proof of a contradiction in Peano Arithmetic; which if correct is not that specific to PA but imports itself into much weaker systems. I'm skeptical of the proof but haven't had the time to look at it in detail. There seem to be two possible weakpoints in his approach. His approach is to construct a system Q_0^* which looks almost but not quite a fragment of PA and then show that PA both proves this system's consistency and proves its inconsistency.
First, he may be mis-applying the Hilbert-Ackermann theorem-when it applies is highly technical and can be subtle. I don't know enough to comment on that in detail. The second issue is that in trying to show that he can use finitary methods to show there's a contradiction in Q_0^* he may have proven something closer to Q_0^* being omega-inconsistent. Right now, I'm extremely skeptical of this result.
If anyone is going to find an actual contradiction in PA or ZFC it would probably be Nelson. There some clearly interesting material here such as using a formalization of the surprise examiation/unexpected hanging to get a new proof of of Godel's Second Incompleteness Theorem. The exact conditions which this version of Godel's theorem applies may be different from the conditions under which the standard theorem can be proven.