As mentioned in https://bitbucket.org/site/master/issues/12308/rest-api-wiki-update-doesnt-work#, the REST API wiki PUT method has worked, but now seemingly returns an error which seems to lead to the fact that it's trying to create a new page with the same name instead of updating the specified wiki page.
After executing the following command through SSH;
curl --user user:pass https://api.bitbucket.org/1.0/repositories/repo/project/wiki/Project --data "data=content&path=/Project.wiki" -v -X PUT
The following error occurs;
<ul class="errorlist"><li>path<ul class="errorlist"><li>Please choose a new title, Project.md aleady exists.</li></ul></li></ul>