Tilde expansion only works in the beginning of a word. (At least in bash, see e.g. https://www.gnu.org/software/bash/manual/html_node/Tilde-Expansion.html )
You could use, say --something=$HOME/filename if you only need to refer to the current user's home directory. Since --output-document takes a mandatory argument, writing the argument in a separate option works too, but for optional arguments it doesn't seem to work (*), so you probably want to use the --foo=bar format usually.
If the context is documentation, it might be useful to at least mention the corresponding short options too since they are shorter to write and your readers might stumble upon them in code written by others anyway. Especially for common options like wget -O.
(* For example, try ls --color=auto vs ls --color auto. I said "seem to", since the most direct mention of this I can find is the man page of the command line program getopt, the library documentation isn't as clear.)