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As I imagine it's php rather than html then you probably haven't over-written the old one and all your old site content should still be on the server, just not accessible because you no longer access the html homepage by your domain.
How do you upload your data? If using a conventional FTP program you should be able to see the files.
If you asked me what happens if you have an index.php and an index.html page I wouldn't know - but it looks like you've answered it yourself, the domain defaults to the index.php page.
I'm not sure what the best workaround would be. I'd imagine there's some kind of re-direct thingy you could do. "301" sounds familiar but no idea how to do one.
Alternatively delete Wordpress and reinstall on a subdomain, transfer everything to it and then copy over to the main directory - but that depends on how Wordpress works. You may then have to manually change lots of links, though hopefully it uses "/" rather than your domain with internal links.
For example a page called 'page1' should, hopefully, be '/page1.php'
Because then you'd have to change em all again. Might be no great trouble but if it's using links internally in the script it could be a nightmare to move it. So try a Google or something on 'moving a Wordpress site', see what's involved?
You'll also need, if going this route, a hosting provider that allows subdomains. Some do, some don't, some charge a small fee.
Another alternative is to creat a new directory, which is basically the same thing but looks more like
but that has the problem of being spiderable by search engines and can screw up your SERPS a little when changing over, but no biggie if you're just doing it while uploading content in the short term.
But primarily, the problem is Wordpress is a website, not a blog within your site unless you carefully set it up like that.
I still have no idea why a browser is picking index.php instead of index.html but I guess it can't show both and for some reason is going with the newer one.
As I understand it, servers have a list of which files to look for as the index when only the directory name is requested. The list works in order of preference. In this case it sounds like index.php is earlier on the list than index.html and needs to move down below it. But if this is shared hosting rather than dedicated, that change could affect all domains on the server.
However, it seems to me that having index.php higher on the list than index.html is pretty unusual. How about going in via FTP and seeing if the index.html is still there? It may have been deleted during the install, so replacing it could fix everything.