If your foundations don't encroach beyond the boundary, there's no need to serve a Notice. Since 2018 nine judges have determined that your common law rights continue if you don't wish to exercise rights under the Party Wall Act.
And even in the days when it was thought that the Act had extinguished common law rights, the idea behind the s1 Notice was simply to give the adjoining owner warning of the intended works, and so no 'dispute resolution procedure' would be triggered.