In my experience if you ask for an extension for a TMA you will always get one; I've occasionally asked for a couple of days and got a week. With EMAs the deadline is sacrosanct. I've asked on the day and a week back; I guess the earlier you see the problem the better. It depends on the problem. Never really meeting my online tutor it took me too many modules before I'd pick up the phone, yet nothing has ever worked better than a short chat. If your request for an extension is genuine, like the kind made of hair, you'll want and probably need a helping hand so take what is on offer.
The problem with an extension is that it could simply be delaying the pain ... you continue in the same vein, blocked, muddled, reluctant to start, or finish or submit; or you haven't done the work and know it.
If you've done the work but are having a mental or academic block then ask here, ask in your student forum (too often quiet) and most certainly talk to your tutor. Often some focused advice will say you are on the right track, press on, talk it through, now write it down and send it in, or that you are trying to write a chapter rather than an essay, to narrow it down, to think strategically.
If you haven't done the work then use the extension to figure out how you can put in enough hours, strategically, to cover enough of the ground and do it. If that means a few late nights or early mornings that so be it.
And shit happens - I've lost parents. The worst. And I've been made redundant. In fact both happened together in my first module and I quit ... I sulked instead of talking to someone. The OU would have been understanding. Anyway, you may have to drop everything for a week or more. Everyone will he hugely sympathetic.
Otherwise, like teachers with decades of experience, the tutors will smell a rat if you simply haven't done the work and don't intend to. If this is the case then personally I'd just knock off a submission and send it in. You may still pass. You may be asked to resubmit. The assumption will be that you've done your best even if you haven't. Or perhaps you have? Honesty is always heard with sympathy.
I'm reminding myself that however distant distance learning might be you are always a phone call away from your tutor. I always regret not taking up the offer of talking it through. They won't think you're an idiot; they'll admire you for caring about wanting to try to get it right. In seven modules I think only once did our tutor group have such an active online presence that we could sort out each others' problems, another the tutor didn't just say they were available, but they proved it by being online moderating and 'seeding' conversations rather than waiting in the background (or just absent). And then a time when I kind of buddied up: she was great at one kind of thing, I could help with another - she was an 'adapter' I was the 'innovator' I guess.
Have I had a reprieve? Not exactly. I have, or thought I had a 10th September deadline - actually thats for resubmissions for April !? So I have until the 24th September. I have worked my way into a corner so will take a day off, then see what I can write in exam conditions, see if a fresh approach can produce something more fluid. It's a relief to be able to step away from a disaster.
I am so envious of people who turn mountains into molehills. I turn mountains into a range of mountains and usually get lost until someone drags me out. TMAs are foothills, EMAs the one mountain. They all count, so press on.