Prepare a new version of the old desktop app. That version shows the user a popup on every app start to update to the new desktop app. If the user accepts, it installs the new app and uninstalls the old one, and could also clean up the old data directories. The new multiplatform app could be bundled inside this update, or be downloaded at that time.
If we need a simpler option, the popup could just download the new app into the users transfers folder, and we could delegate to the user installing the new app and uninstalling the old app.