Mehhhhhhh, the name change isn't worth doing mass edits over. Nor is the change in URL, since you can click on most old URLs and be taken to the right page on the new site. There are many thousands of posts that link to oxforddictionaries.com (more specifically ~11k) and most of these links are still good and don't really need editing, since they just redirect to the current lexico.com site. Most.
For non-critical issues like this, I fix them when I fix other issues on the same page, or if the question is already bumped and on the home page.
Some links have been broken by this change, which is a more critical issue, so it might be a good idea to do some active editing, as long as we don't flood the home page. Right now, I see that the change has broken many (almost all?) of the blog links. I don't know if they're planning on reposting these articles on the new site, so I'm preparing for the worst. Not all of these URLs were saved in either of the archiving services I know of, so I've been saving Google's cached version of the page into Archive.org (the cached page is most easily found by searching Archive.is). Unfortunately saving any URLs that have not been backed up needs to be done as soon as possible, since Google caches don't last forever :/ (Yes, my method is complicated, but I do it this way for a reason.)
And then there are some links to the regular dictionary entries that were broken a while ago by a different URL change, which can be found here. Plus, some URLs were also broken when another one of Oxford's dictionaries (the OALD) changed URLs; a list of posts affected by this can be found here. While I'm on the subject, I'll also note that there are old broken Google NGrams links/images in these posts. I've been fixing these every now and then for a while.
With the new Lexico site, another thing I noticed is that there is now only one dictionary instead of two ("Dictionary" and "Dictionary (US)", switched via the dropdown at the top of the page). This really isn't as much of a loss as it sounds, since both dictionaries were very close in content (see links above). Therefore I would consider this another non-critical issue for most posts.
There are also a couple of questions about dictionaries, especially here on meta, that could use updating (even if the links aren't technically broken) such as: