It's a risky maneuver, but when I weighed the chance of my professor getting upset/peeved with the chance of my apps not getting in by my personal deadline, I decided it was worth the risk:
I emailed my last prof and told him flatly that everything was in, except his LOR. I'd taken the LSAT, the other LOR was in, my app was submitted. I explained, kindly, that I understand he has a lot to do these days, but that after my LSAT score comes in the only thing that will be holding up my application is his one remaining LOR.
He mailed it about 1.5weeks later. He didn't seem upset about my email at all, either. But I do think that if I had not laid it all out on the table like that, I'd still be waiting.