I have filed many dozens of LNFs, and by now I have received replies at every hour of day and night. (I should clarify that I've always filed LNFs during the week, so I have no expereince with LNF claims made on weekends.)

I don't know where they are, but they seem able to process them within 24 hours always in my experience.

But as I suggested somewhere above: Make sure you make the claim at an hour of the day such that you can check your email right before the same time the next day, in case they took almost 24 hours. (The timing of all this is especially important if you reserved a "non-cancellable except within 24 hours" rate at marriott.com.)

There's nothing special about midnight. The deadline is 24 hours from you file the cliaim. You file the claim at 3 am, the deadline is the next day at 3 am. You file the claim at 7 pm, the deadline is the next day at 7 pm. Very straightforward most of the year. (Exception: when either the USA or where you are are shifting between standard and daylight savings time, and then "24 hours" is a bit less clear to calculate. But that's no more than 4 days a year -- 2 days for the USA, 2 days for your country if it's different.)