Changing your WP theme is like changing your clothes. It's the same person just in a different outfit. You're still as healthy as you were before and still have the same facial appearance. Maybe because you're now dressed badly you will might be ignored, but you would still as easily be found because you're business cards and contact details. This is like changing a theme, in that you can be found but once found might be ignored because your new theme is not as good. Alternatively you new theme might be better and some new people might be drawn to you, likewise some of your old customer might feel intimidated and leave you altogether.
There should be no issue in updating your theme from one to another, providing your SEO plugins remain the same and you have the same data in your web pages. Sometimes a change of theme is required, particularly if a theme is giving you issues. I remember a while back I ditched a WP theme because it was complicated.
Some great themes include SEO properties in them. I would copy and paste these properties in an independant S.E.O. plugin. I like to use S.E.O. Yoast, and from there change your theme without an issue. For security purposes you can use something like 'Maintenance Mode' so you can make the transition easy. If you're struggling with the plugin you can change the .htaccess file to ban all IPs but your own whilst working on your website.
Hope this helps.