The oath itself wouldn't stop the British Parliament passing an Act of Parliament to abolish the Crown and replace it with some other system. In theory, the monarch could refuse to give their assent to the proposed law but given that would cause a constitutional crisis, in reality the chances are the monarch would assent and the system could be changed.
It seems it would take a republican government in power _or_ huge public demand that the monarchy to be abolished for that to happen which seems unlikely any time soon assuming King Charles III and his successors don't err massively.