A TRACON Controller reported aircraft on approach were initiating go-arounds due to wind shear. A regional jet climbed uncontrollably 3;000 ft. above its assigned altitude.

Narrative

I was working ZZZ and North arrival. I had been on position for about 20 minutes and there was weather on the scope when I sat down. I was briefed that there had been some deviations but just light chop. And area of heavy precipitation which had been west of ZZZ1 eventually moved onto the final but aircraft were still taking the approach and landing. Eventually we got some LLWS alerts which I issued which in short time were followed by several go arounds and a microburst alert which I also issued as well as described the precipitation along the final. Aircraft X said he needed to discontinue the approach so I issued a 220 heading and 5000 feet since departure was getting a go-around in front of him and I couldn't turn him left or right due to airspace constraints. I yelled to departure and shipped him. In the mean time every other aircraft on final clogged up the frequency saying they also wouldn't take the approach. At some point Aircraft X said that he encountered almost unrecoverable conditions and had climbed to nearly 8000 feet.We need to have a much better plan of action when there is weather in the area. Our current method of giving the approach until aircraft refuse to take it while informing them of the weather will only result in this kind of situation in the future; which I find unacceptable. There was already a ground delay program in place when this happened; but there were no miles in trail out and I was oversaturated with aircraft. In the future I will will make a bigger point to not run aircraft through a final that is heavy precipitation even if outside they are below the layer and willing to take it. Also I think pilots need to be more vocal about saying what they see outside because often times we have no way of knowing what the weather is really like outside.