Last term I taught a large undergraduate course. I had assignments (online) due a week after we finish a chapter. There were two minor problems with this. The first was that I had to deal with students who enrolled one, two or even three weeks into the term and gave them extensions of the first several assignments, which involved logging into the website, finding the student, setting his/her new deadline and answering the student email. Secondly for some of the chapters that take more time than others, some students would bombard me with requests for extensions. So I am thinking of adopting a different method next time. Specifically, I plan to divide the 12 chapters into 4 modules with 3 chapters in each. Each module has a deadline for all 3 chapter assignments. I am hoping this would give students more flexibility on their time management, and have a little more time working on those long assignments. It also saves me from handling those extension requests because of late registration or others. The potential peril I can think of is that some students may not start on the assignments until the last minutes and it would be very difficult to crash 3 assignments within a short time. I would appreciate other thoughts on the pros and cons of this method.
I would recommend not batching up assignments to a common deadline, because that makes it even more challenging for students to manage their time properly. If someone waited to the last minute on an assignment in the prior regime, now they'll be waiting until the last minute on 3 assignments and be in even worse shape -- compounded by the mental effort in prioritizing which one they should work on first. Under the original regime, they have more opportunities to engage in this cycle and get feedback on how their study skills work for them.
My response to these types of problems is to have a blanket policy of dropping a certain number of grades from consideration; for example, in my classes, there are 12 weekly online quizzes, and the lowest 3 are dropped for each student at the end of the course. There are absolutely no makeups or past-due work allowed for any reason, which massively clears out my own time (absolutely no chasing students or managing late work after my grading process is completed). Plus, the "drop N grades" is automatically supported by the Blackboard management system.
In my classes, I don't give any special allowances to students who first show up later in the semester (but this may vary for your institution). I had a student join my class in the 2nd meeting last week, and he had already missed the first quiz submission; he asked "Can I make that up?", and the answer was simply, "No", and being pointed to the syllabus for the drop allowance, which in the years I've been using it has always been accepted quite gracefully. In my opinion, it's the most efficient use of your time as an academic.
In your case, I'd try a larger number of smaller assignments, spreading the work out rather than bunching it up. Your grading load doesn't change, but the students can manage their time more easily. Put "due means due" in the syllabus and set a hard cut-off in the learning management system.
No matter what you do, the poor students will wait until the last minute, turn in rushed work, and earn poor grades. Like you, I have a desire to help such students, but I haven't found anything that works. By having more assignments, each one is worth less, and perhaps some students will learn after the first couple of rushed assignments earn bad grades.
For students registered late, which is really a different problem, set a time from registration date by which they must be caught up, put it in the syllabus, and set the LMS to enforce it, which you can probably do easily on an individual basis.
Be sure to see my comment on "late days." It's a comment to the original question.
As a current engineering student, third year, I would be rather wary of the idea of grouping all the assignments together under a common due date. Especially if, the course in which this takes place is a first or second year course. This is because younger students often manage their time rather poorly. As Daniel R. Collins noted, attempting 3 assignments the night before their due date can be a discouraging experience and yield dismal results. I would also add that, as a Prof, you should set the timetable and not the students. It is very annoying to see a lenient teacher extend a deadline because a student is not on time. Especially so, if said student is behind because they are not working on the assignment and is using their time for other activities. After all, part of the skills one is to develop through a bachelor degree is time management. Lastly, you should check either with your faculty or department if they have a late work policy, if so you should reference it in the syllabus, if not get them to establish one.
I've tried both methods for our introductory classes. To be honest, I haven't seen very much difference. The students who will wait until the end to do all three sections are going to be the ones that will wait until the end to do each individual section. In all cases, you're getting rushed work that they're likely to hand in late, except that they're spreading their work out (better learning, mayhaps) but sending you more e-mails, or they're cramming their work but sending you less e-mails.
To help them with time management, you could give them estimated completion times. Even if they're much faster or slower than the estimate, they can figure the others out relatively speaking after doing one or two ("Professor said section 2 takes two hours and section 3 takes four hours. I did section 2 in only an hour so I can probably do section 3 in two hours").
To avoid the e-mail bombardment, I'd consider instead just instituting a general late policy. It could be one that slowly penalizes more (so that one day late = 10/20% off, two days = 20/40%), or a one-off decrease (50% off), but they key is to balance it: too punitive, and they'll still e-mail you, too easy, and they'll take too much advantage of it. Or you could say "I will permit one module to be handed in up to X days late".
As others have already given good answers for the 1-3 week late admissions problem (dropping lowest grades and/or a blanket reduction of points for late work). I will offer another option for dealing with the problem of too many extension requests for homework assignments when dealing with online courses.
A policy which has seemed to work well during my studies was one where professors in common undergraduate requirement courses agreed on alternate due dates that mimicked the due dates of on campus courses or simply set their due dates to match on campus meetings of the same or a similar class. It seemed to help new students budget their time more effectively - keeping all the assignments from each class being due on the same day (Sunday) - and did not seem like a lenient policy as it was essentially the same amount of time to work on assignments as it would be for on campus students (may get an extra day or so depending on when one posts the assignments online).