The origins of writing in Ancient Egypt are not very well known. Around 3000BC, the hieroglyphs appear, and the system seems to be fully formed at that time. This might be due to a lack of documents. Another hypothesis would be that the Egyptians took the idea of writing from the Sumerian, but this is unsure.
Anyway, the hieroglyphic signs themselves are very different from their cuneiform counterpart, and use pictures of typical nilotic fauna and flora, so most of the system must have been developed in Egypt proper.