correlation coefficient between weather data and power consumption
I am not very sure how effective NN would be for this problem. The way I see it is that you have 48 entries in a time series and now you are trying to predict the next 4. Keeping aside the correlations for a second, it seems like there is very little data to "learn" from. I mean the power consumption is a seasonal thing (much like temperature) but you have only 1 years data so it should be difficult for NN to capture this seasonality. 

I would like to raise another point about the way you are using correlations. If you are taking correlations using the actual temperature value then I do not think that to be a good idea. Think of it in this way, I would be using a lot of power at both very high and very low temperature. So instead of using the raw temperature values, I should normalize them with respect to some "comfortable" temperature value by taking the absolute difference between the actual temperature and the "comfortable" temperature. 

For instance, let us say that "comfortable" temperature is 25C, then both 5C and 45C would be normalized to 20 and this looks more plausible as my power consumption should be high in both the cases. This might also explain the poor correlation that you observe for 3 months of data. What should be the "comfortable" temperature is an entirely different story altogether. 

Do let me know if this line of thought makes sense.