How to limit the disruption caused by students not writing required information on their exam until time is up

Even despite repeated reminders, exams can be a tense time. Some students may honestly forget, and it would be overly harsh to deny them their exam for this. OTOH, some students may do this deliberately as a way to game the system for a little more time, and that should be discouraged.

I'm not sure you have the authority to completely address this on your own, but perhaps you can document it in a way that helps distinguish exam nerves from gaming and nudges students towards following the rules. Something like this maybe:

Take two boxes, visibly distinct. Students who finish on time put their papers in Box A and leave. Students who don't finish on time put their papers in Box B (ETA clarification: after they have added their details, don't want to lose identification of papers), and are required to fill in an extra form that gives their name and explains why they didn't fill in their details ahead of time. (Give them this form at the point where you collect their exam, so you're not giving them an opportunity for even more writing time!)

You then have a record of who is doing this, and if you're invigilating the same students repeatedly (or if you can persuade other invigilators to do the same) you can distinguish between one-offs and those who make a habit of it. You then have ammunition to go to your course head/etc. and say "this is a problem, we need to address it". (Or not, if it turns out you don't have repeat offenders.)

From their side, the fact of being singled out and knowing that it's being recorded gives a bit of a behavioural nudge. It tells them this behaviour isn't within the accepted norms, it gives them the impression there might be consequences if it persists, and it also shows other students that this might not be a great idea to copy.

In a similar situation, I have held time at the beginning of the exam to fill out the needed information. Once everyone is there, tell them to start writing their names on the test. You can see them all paging through the test quickly. Have them close their test and put their pencils down when they're done, and let everyone start the test when everyone's done.

I haven't done this with 100s of students, but it can be fairly obvious when someone is actually working not just writing their name (e.g. poking at a calculator).

You can also hand out only an answer sheet, and hold time for everyone to fill it out, then pass out the tests. (This is in the linked question.) This also is handy for reducing number of pages printed, especially if color costs more, sometimes.

Finally, tell them why you need it on every page. If it will slow you down grading (and getting them back to students), tell them! As Geoffrey's answer points out, they're not being malicious, probably just forgetting.

At the end of the exam time, tell everyone "pencils down".

Once everyone is still with pencils down, announce "You have 1 minute to write your name (and other information) on each and every page of your exam. If your name and class isn't on your exam at the end of this minute, your exam will still be collected without your name on it, and you will not be graded. This is your last chance. Start now."

Then wait 1 minute. They state "pencils down" again.

Now collect. Anyone who doesn't have their name and class written literally had a minute to do nothing but that, with consequences spelled out, and time dedicated to only this task.

That 1 minute should save you more than a minute. It is plausible that someone might spend that one minute doing a tiny change to an answer they worked out in the gap, but it isn't going to have much impact.

This is different than warning them 10 minutes before the end, because at 10 minutes before the end they still nominally can do work, and are focused on the exam (and possibly barely hearing you).

The goal is to parallelize the "fix" period, instead of making it sequential, and giving students who have forgotten to do it a sanctioned period to do it in.

A student who realized their mistake at "pencils down" might not feel comfortable opening the exam to write their name without your explicit permission, as that is viewed as cheating. So rather than giving permission individually, you give it to everyone at once.

(Feel free to modify the "1 minute" based on your experience of how long it takes to actually write your information on each page.)