Professor didn't write recommendation letter on time, what should I do?
It sounds to me like you're in the US and I'm going to answer this from a UK perspective, but I expect the situations are broadly the same. I'll first discuss a similar thing that happened to me, and then give some advice.

So, this exact situation happened to me last year with two different universities, and two different letter writers. All the places I applied to required a minimum of two reference letters, so in each case, I was down 50% on letters.

In the first case, the university notified me that I was missing a reference letter and I had time to find someone else who was willing to very quickly write me another letter. My application there was ultimately unsuccessful, but not because of this (it's the best uni in my country and my application was wildly optimistic).

In the second case, the university didn't notify me at all that they had only received one reference letter in support of my application. In fact, I didn't find out until they mentioned it in the interview, after which they subsequently gave me an offer, still only having seen one reference letter. So it didn't matter in this case either.

My advice to you is the following: contact the schools which are missing letters. Ask if it matters that your letter might be arriving slightly late-- I expect it probably won't. People working in academia are aware of the time constraints that academics have and will likely be lenient in this kind of situation. I also don't think it will reflect badly on you, as you have done what you can to rectify the problem.

Secondly, send another gentle reminder to the letter writer in question, or, if possible, pay them a visit in person. Chances are, it was an honest mistake and they just forgot. 

Good luck with the rest of the applications!