Some children need more sleep than others. It really is as simple as that. If he's resting for a full hour then it might well be that he's had sufficient sleep.

While I do understand that this is your break, the expectation of a 2 hour break is high. Most adults have 30-60 mins lunch break and fortunately you have got that. There are times in this business that we have non-nappers and this might be one of them.

I personally have a 2 hour rest period in my day home and all children are expected to rest however if someone wakes up before the end of the rest time, then I do permit them to come and play quietly. While of course that is an interruption of my time, it's better that those who actually need to sleep are allowed to. I'd rather have a shorter break than an afternoon of over tired children whose rest was poor.

The other option is to make morning's much busier so he's actually tired enough to rest for the time you are hoping to have to yourself. Having a busy outdoor time during the morning is the best way of achieving this.

Other than that you are going to have to get used to his upset and hope that in a short span of time, leaving him to self settle when he wakes will work. It's unrealistic to expect an awake 15 month old lays quietly so either you are going to have to train him to go back to sleep when he wakes or you are going to have to get used to the noise, or you are going to have to let him get up and play. Personally, I'm not okay with leaving a child who to cry for a whole hour especially at that age.