Probability Calibration : role of hidden layer in Neural Network
It makes sense that a network with ReLU activations produces worse probabilities than a sigmoid activation when you think about how these activation functions interact with the unit in the output layer of the network. Sigmoid scales the output to be in the range $[0, 1]$, so the values going into the final sigmoid classification unit will be fairly low. Meanwhile, the ReLU-based network has the potential for very high activation magnitudes, which will skew the estimated probabilities to either end of the spectrum.

Having said that, calibration of neural networks is not very well understood in general. A paper was published last year at ICML (https://arxiv.org/pdf/1706.04599.pdf) which showed that often neural nets are poorly calibrated, but it doesn't really explain why this is the case. However, they did investigate factors that affect calibration, and show that deeper, wider networks can often be less well calibrated than their shallower, narrower counterparts. They also show that applying $L_2$ regularisation can improve calibration.

Given this information, I'm surprised that your (fairly shallow and narrow) network is so poorly calibrated. It might be overfitted to NLL, so I would try applying heavier regularisation or training for fewer epochs. If this doesn't work, try implementing temperature scaling, which is described in the paper I linked above. They provide a PyTorch implementation here, so it shouldn't be too difficult hopefully to translate into Keras.