I can't speak to urgency, because that could be interpreted as a future looking timeframe.
As a process, I would enable the Swift 5 stricter concurrency checking in code that you control to gauge what the important issues in your code base are, and decide what it would mean to fix them in terms of complexity. After that, try out the Swift 6 language mode, and see what other issues the complier identifies, once again with an eye for complexity. You can then take that information about the technical complexity and use that as information in your overall scheduling and mangement of your software project, and when the best time to do the work is.
Since you can do that on a module-by-module basis, there's no need to jump to Swift 6 all at once. You can turn it on just for one module where perhaps there are no issues, or they are easy to address, after updating the language mode, and expand outward from there. And looking at this from the other direction, if there's a module where the issues are extensive in order to move forward, perhaps doing the work on that module is the higher priority because it will take longer to complete. Those are the kinds of choices you'll have to think though based on the exact specifics of your software project.
The above pertains to your code. The other aspect here is code you're not in control of, either from third-party libraries, or even other teams within your company. You certainly want all of your dependencies to be aware of Swift 6 (i.e. you're not depending on a library that is not actively maintained) and charting their own path to adopting Swift 6 in time. This could be as simple as updating to a newer version where the work is already done. But as I said above, the language modes are meant to provide time for the migration — library vendors may need more time to migrate than app-level code, as they have the needs of many clients of their API to consider.
Again, I realize I'm not answering your question directly, but I hope the above is a useful strategy to inform your thinking about the problem, and how you can incrementally move parts of your app forward over time as it makes sense for each module to do so.
— Ed Ford, DTS Engineer