I just went through this - my strongest recommender, asked in September, said he would write a very strong letter, I've done research with this prof., currently work as a research assistant for him (i.e. see him 3+ days per week), well known in the field, etc. I sent him a few reminders between when I sent out the prompts via applications (around Halloween) and the week of my first deadline. He kept saying "don't worry" I'll probably be a day or 2 late submitting them but adcoms don't hold recommenders to the same deadline as applicants and it won't affect your chances. Fast forward to deadline day, no LOR uploaded - sent reminder email, rinse and repeat. End result: he was 2+ weeks late submitting most of my letters (6/8 of my deadlines were 12/1 or before). I know people that got called for interviews before my recommender even uploaded his letter. Even though people will tell you not to worry about it I would stay on top of this, especially because it's a week of holidays and then new years.