Code snippet for update script to use new theme logo

custom-logo-update-script.php

<?php
/**
 * Theme Update Script
 *
 * Runs if version number saved in theme_mod "version" doesn't match current theme version.
 */
function prefix_update_check() {

	$ver = get_theme_mod( 'version', false );

	// Return if update has already been run
	if ( version_compare( $ver, '2.0.0' ) >= 0 ) {
		return;
	}

	// If a logo has been set previously, update to use logo feature introduced in WordPress 4.5
	if ( function_exists( 'the_custom_logo' ) && get_theme_mod( 'logo', false ) ) {

		// Since previous logo was stored a URL, convert it to an attachment ID
		$logo = attachment_url_to_postid( get_theme_mod( 'logo' ) );
		if ( is_int( $logo ) ) {
			set_theme_mod( 'custom_logo', attachment_url_to_postid( get_theme_mod( 'logo' ) ) );
		}
		remove_theme_mod( 'logo' );
	}

  // Update to match your current theme version
	set_theme_mod( 'version', '2.0.0' );

}
add_action( 'after_setup_theme', 'prefix_update_check' );

Here is a variation that avoids the use of a theme version check, instead just checking for the existence of the custom_logo mod: https://gist.github.com/ericnicolaas/fa64a99ef02f9efbcdf89345c74d6aff