Log delay in Amazon S3
What you describe (log files being made available with delays and being in unpredictable order) is exactly what is declared by AWS as behaviour to expect. This is by nature of distributed system, AWS S3 is using to provide S3 service, the same request may be served each time from different server - I have seen 5 different IP addresses being provided for publishing.
So the only solution is: accept the delay, see the delay you experience and add some extra time and learn living with this total delay (I would expect something like 30 to 60 minutes, but statistics could tell more).
If you need log records ordered, you have either sort them yourself, or search for some log processing solutions - I have seen some applications being offered exactly for this purpose.
In case, you really need to get your log file with very short delay, you have to make the logs yourself and this means, you have to write and run some frontend, which gives access to your files on S3 and at the same time keeps logging as needed.
I run such a solution, users get user name and password and url of my frontend. As they send the request, I evaluate, if they provide proper credentials and if they are allowed to see given resource, and if so, I create few minutes valid temporary url for that resource and redirect the request to that.
But such a fronted costs money (you have to run your frontend somewhere) and is less robust, then accessing directly the AWS S3.
Good luck, Lulu.