How to deal with incommunicable co-author

When dealing with extremely busy people, you want to formulate your emails in a way that makes it as easy as possible for them to give you what you need.

Providing one's opinion on a paper is a complex, open-ended process. Even more so if you are a coauthor - you need to not only read and think about what is written, but also consider what else could have been written to make the paper busy. In a situation as you describe, I'd consider it perfectly normal for the professor to take a few months to get back you even they seriously intend to do so. Thus, your request is not a good choice.

Instead provide them with a straight-forward way out that works for you. Thus, complete the draft and makes sure it is completely polished. Make a plan regarding what you want to do with it. Then send it to the professor, pointing out that you and B agree that the paper is finished, and that you want to submit it to the arXiv/Journal X, and that you just ask for their consent to proceed. If the contribution of the professor is very limited, you can consider also stating merely acknowleding them as an option. If C or anyone else respected by A has read the draft and also considers it ready, include this information. Overall, you want to make it very easy for A to just respond "its fine, go ahead".

If this doesn't work, there is one way to escalate further, but I'd consider this borderline from an ethical perspective. That is to let A know that if they don't object within eg 2 weeks, you will consider this as permission to go ahead with the plan. Only ever think about this if your draft is really, really done.

Rather than asking A for specifics on the paper, ask them for advice on how to proceed. They may just tell you to go ahead, listing them as co-author.

You need their permission to proceed, but you don't need their further participation. If necessary, stress your need to get something out there for your own career(s).

But if you make their task harder than they are willing to work, then you will possibly stay stalled for a long time.

If they give permission to continue with their ideas included just, as a courtesy, send them updates as you go along.

You have to include them to avoid plagiarism in this case and, including them, their permission to publish is needed. A reputable publisher will probably require this, in any case.