This is by far the best answer to the question, because it addresses it from both sides

I think you should inform the other journal directly in your review. "Below I include a modified version of my review of this manuscript at a previous journal. Most of my comments were not addressed although issue X was corrected" This is actually important to the editor, it is valuable information. I am an editor of a journal and I would prefer if a reviewer disclosed this. But ultimately its up to you given you are giving your labour for free.

I strongly agree with the appendix option in most cases. In the main text, state the modified theorem. Then something say something like, "To prove this theorem one can take the strategy in X used to prove a similar theorem, with modifications Y and Z (see appendix for a full proof)." This also means reviewers will read it, and can tell you to add it back to the main paper if you need to. If you omit it entirely they won't peer review the proof and this can slow things down if they want modifications to the proof in the next round

The Australian system is closer to the British system. A bachelors is 3 years. Honours is an additional 1-1.5 years which is generally 40-100% research, culminating in a thesis, which is typically one academic paper in length but a bit more verbose and not as polished. A PhD is then 3.5-4 years. An honours is similar to a masters or the first 1-2 years of a PhD program in the US. It's not 100% analogous but this can give you an Idea. I'm a TT academic in Aus and did my PhD in the US

I would usually provide several examples 3-4 on the first pg and then say fix these types of errors elsewhere. There is always a happy medium.

I really think that a lot of the answers/comments here while correct as written, are really missing the mark in flavor, it depends on how you write that email. I have expanded in my answer below. The focus should be on collaboration and shared interest, do not point out the omission, the authors can infer that on their own, its their judgment to make.

This is the best answer. Another reason why deleting the acknowledgement section entirely is better than other answers (suggesting the OP thanks their fellow students, family and friends) is that by thanking others and not your advisor it reads like a deliberate choice, and draws attention to the omission, whereas if you don't have the section many people probably won't even notice, and if they do it just looks like you didn't want to write that section and are bucking tradition, which will raise far less red flags.

@buffy actually the OP has described nothing about the interaction between A and B at all. Both your answer and my comment are the same level of speculative, but at least my comment is a conditional statement: if X then make sure Y. Since the OP has said nothing about whether Y is true, it is fair to recommend they make sure they have done their due diligence to make sure Y is the case. It is also speculative to assume B had not benefited at all from intellectual knowledge from A, even if indirectly through the supervisor. All I am doing is making the assumptions explicit.

If B insists on sole authorship, B must not include any conceptual or writing contribution from A in their paper. B learned about the problem from A, and it would be easy to accidentally include an insight from A in the paper that B assumes is trivial or well-known, but is actually novel, and an intellectual contribution warranting co-authorship. Given A has worked on this problem for so long, this is quite likely. The safest option for B is to offer co-authorship. This also will lead to a faster publication because A is probably more familiar with the literature than B.