Can you code any HTML? Do you have FTP access to the server? If it were me, I’d just create a stand-alone static HTML page named index.html then place it in the http root (www) directory. WordPress sometimes makes easy things much more difficult than they should be.
If you’re saying that you put up a coming soon page and it doesn’t show up when you do a Google search, there could be a number of issues. First and foremost, it could be that Google hasn’t indexed your site, yet. Second, it could be that it did index your site and the site isn’t ranking well. For this items, you need to consider how long the site has been up and how good your SEO is. Side note, why do you want an under construction page to show up in a Google search if you have nothing to show?
Maybe you’re saying your site isn’t showing up when you Google atjgraphics — or whatever your business name and domain happen to be. I’d go back to how long has the site been up and has Google indexed it yet.
What I would suggest doing is put the site you’re working on in a subdomain and then having a simple under construction page.This could be a WP site or it could be a simple index.html doc like @Just-B suggested.
Lastly, I know there are under construction and maintenance mode plugins. I can’t really comment on them as I have no experience with those. They might be the solution you’re looking for without having to upload your site to a subdomain and then move it.