Sorry that you've had such a tricky day 
Just to reiterate what previous posters have said, your son may be using so much effort holding it together at nursery that when he gets home with you he just feels he can let it all out because he can trust you. Don't see it as a judgement on your parenting - we all go through phases of beating ourselves up about this, I do it when my son's alternative provision tell me he is never violent/aggressive there! - but it's far more likely to be an indication that he feels secure and safe with you and is 'masking' or holding it together at great effort when he is with others.
Maybe when he gets back from nursery he would benefit from some 'down time' or some specific sensory input to help him let out some of the tension that has been building? Different things work for different children, depending a bit on their sensory needs and personalities, but things that have worked (sometimes) for my son include - crunchy snacks (crackers/popcorn/carrot/apple/ice cubes etc), physical exertion (scooting home, swinging on his climbing frame, digging in the garden) or down time (audio book in his bedroom with his fav activity - pushing cars/trains about - and no expectations on him to interact). Perhaps having a bit of time apart from his sister when he first gets home so he can sort of level himself out without having to also manage interacting with her? (I appreciate some of these ideas may not work logistically for you, just pick out anything that you think might be useful!)
Re hurting his sister - perhaps you could start jotting down in a notebook/on your phone when it happens - the time, what was happening beforehand, and then what happened. Also maybe what happened after, including how you dealt with it. I find it really helps me to spot what is triggering my son if I write down several incidents - sometimes just the act of stopping and writing it down makes me aware of something, or after several incidents I might spot a pattern.
Importantly, remember that tomorrow is another day, and that you are a fabulous mum who clearly loves her children x