I have migrated several web sites from HTML to WordPress. My experience was good using the following approach:
- Install WordPress in a subdirectory and get it working fully there;
- rename the HTML site's index.html/.htm file and make WordPress look like it is running from the root using these instructions http://codex.wordpress.org/Giving_WordPress_Its_Own_Directory#Using_a_pre-existing_subdirectory_install
In your situation, I would (and have successfully done this) move the old site, including domain name, to your new hosting FIRST, before installing WordPress.