Obvious solution: don’t switch on the live site. You could use a staging environment instead, but this can become tricky with ecommerce sites if orders have been placed in the meantime. Alternatively shut down your site for a couple of minutes.
If this is not feasable, there may be other ways. To understand, how theme switching works, check out the switch_themes( $stylesheet ) function in wp-includes/theme.php. This is the function that manages transition from one theme to another.
I see many ways that you could inject your custom code in there somewhere, you’ll have to test which way fits your needs best.
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Listen to
update_option( 'theme_switch_menu_locations', ...change. This is probably one of the first ways you could inject custom code in. However, here you do not really know the the new template -
Listen to
update_option( 'stylesheet', $stylesheet ), here you know the new theme ($stylesheet) and can get the currently active one viawp_get_theme(). The theme hasn’t switched completely yet, so you can still create the new options as you see fit -
Either listen to
update_option( 'theme_switched', $old_theme->get_stylesheet() )or todo_action( 'switch_theme', $new_name, $new_theme, $old_theme ). The theme switch for most parts has already happened, default values will have been set and visitors in the meantime will see your “ugly” site.
I would probably go for solution 2. (ok, false, I wouldn’t do these changes on a live site). The new options have not been checked/set yet, so you can copy what you need from the old template and prepare these values. So when WordPress checks for the settings, you’ll already have them pre-filled.