Did you use the Font Manager to set the font settings on the site? If so, please double check if you have set the settings correctly:
In case you have everything setup correctly but the fonts are still not correct. You might need to update all of the URLs on the site to the live site URL.
Please install the Better Search Replace plugin then do the following:
search for http://exampleolddomain.com and replace it with https://exampleonewdomain.com and
search for http:\/\/exampleolddomain.com and replace it with https:\/\/exampleonewdomain.com
Replace the domains above to the previous and the live domain name of your site.