I am currently working on a website for a client who was unsatisfied with the website the previous developer had created. The previous developer however did not hand over the domain which meant we had to put the new website on a new domain. The previous site has been taken down but when I use a search term in google to find the site, the old site is still in the results and shows up higher than the new site even though it is a broken link. In google search console I have uploaded a sitemap and I have followed standard practice for SEO and I still have the issue. I have no contact with the old developer as I have tried and he is unresponsive. What are my options to rectify the issue?
Use Google's Remove Outdated Content tool to notify Google of broken links in their search results. This works whether you own the website or not.
In my experience, Google will remove the broken link from their index within 24-48 hours. Of course, Google does check the link themselves to make sure you're not trying to pull a sneaky.
Google eventually notices and culls broken links on their own, but this is a great way to expedite the process.