No, it will not change automatically. It was declared in 1952, and affirmed in 1960, that the Queen’s children (and therefore her male line descendants) belong to the House of Windsor.
The difference is that the Queen and her children belong to cadet branches of different agnatic houses: the Coburg branch of the House of Wettin in the case of Her Majesty and the Glücksburg branch of the House of Oldenburg in the case of her children. So there is one cognatic House of Windsor in the UK/Commonwealth sense but two agnatic Houses of Windsor in the genealogical sense.
The future Charles III can declare whatever house name he wants if he wants to change it from Windsor.
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