I have just given my ds a pillow a couple of weeks ago - he is nearly 21 months. I was a bit concerned about giving it to him as he is a tummy sleeper too, but he moves around all over the cot and bangs his head on the headboard constantly as well.

He is getting on okay with it - it took him a few nights to get used to it, but I do think that he is sleeping a bit better now that he is.
I know that we gave dd1 a pillow when she was quite a bit younger, but she slept on her back, and I can't remember when we gave it to dd2 (tummy sleeper), but I'm pretty sure she was no more than 18 months.