Best practice is to copy your live site onto a staging site, where you can apply the new theme and make whatever edits you need to, then copy those settings back over to your live site.
You will “lose” your widgets no matter what theme you switch to. If you are using widgets that are coded in your theme itself, those won’t be available to use in any other theme unless you copy them into either a child theme or a custom plugin. Even standard WP Core widgets will “lose” their content – for every theme change, WP assumes you want to start from scratch, so you’ll need to set those up over again. This is another reason it’s helpful to set up a staging site – you can copy and paste rather than trying to remember what content you had where.
The process depends entirely on how much theme customization has been done, and how much customization you want to do on the new theme, which is a matter of preference and also a matter of your site’s unique needs. Theoretically, you could copy everything to a staging site, swap themes, set up widgets and tweak a few things, then copy everything back to production within a few hours. It depends largely on how quickly you’re able to copy sites – which often comes down to your host. Some make it super easy while others are more of a manual process. There are plugins that take backups and help you migrate databases which can also speed up the process if you’re not using a host with built-in staging/cloning capabilities. And depending on your host’s file manager, it may be super quick and easy to copy all the files in your live site folder to a staging site folder, and use phpMyAdmin to download the live database and import it into the staging database.