Updated site ready to replace old site: best way?

My new, mobile-friendly site is ready to replace the existing site, and I’d like to learn the best – not necessarily the fastest or easiest – way to actually make the switch.

The new site was created with a both a new theme and a fresh WordPress installation in a sub-directory on my server. In preparation for making the switch, I’ve installed the new theme into the original WordPress installation in my root directory.

As the new site will consist – at least initially – of only pages (not posts), I’m guessing I would choose to export pages only from the updated site and then import that file to the existing site in the root directory.

I realize I may have some tweaking to do after making the switch, but I just need to be sure that the export/import process will overwrite/change only the old pages and nothing more.

Am I correct as to the process, and is there anything I haven’t considered?

Many thanks!