I.e. unlink the old file and create a new one by the same name (with a backup step in between). So the other hard link would keep pointing to the old version of the file.
Sorry, you're right. Sadly I cannot edit my comment since too long time has passed. I should of course tested with an added hard link. Vim does do the rename/unlink if there only is a single hard link to a file, which makes inotify behave different than expected, so my guess was that Vim would also be OK with breaking hard links. But it makes sense that it wouldn't, since hard links are a much more integral part of Unix.
$ echo 1 > test1
$ inotifywait --quiet --monitor test1 &
[1] 14321
$ vim -c 's/1/2/' -c x test1
$ cat test1
2
$ <no output from inotifywait>
With multiple hard links, the inode is preserved (only difference is adding the call to ln):