$('img.svg').each(function(){
        var $img = jQuery(this);
        var imgID = $img.attr('id');
        var imgClass = $img.attr('class');
        var imgURL = $img.attr('src');

        jQuery.get(imgURL, function(data) {
            // Get the SVG tag, ignore the rest
            var $svg = jQuery(data).find('svg');

            // Add replaced image's ID to the new SVG
            if(typeof imgID !== 'undefined') {
                $svg = $svg.attr('id', imgID);
            }
            // Add replaced image's classes to the new SVG
            if(typeof imgClass !== 'undefined') {
                $svg = $svg.attr('class', imgClass+' replaced-svg');
            }

            // Remove any invalid XML tags as per http://validator.w3.org
            $svg = $svg.removeAttr('xmlns:a');

            // Check if the viewport is set, if the viewport is not set the SVG wont't scale.
            if(!$svg.attr('viewBox') && $svg.attr('height') && $svg.attr('width')) {
                $svg.attr('viewBox', '0 0 ' + $svg.attr('height') + ' ' + $svg.attr('width'))
            }

            // Replace image with new SVG
            $img.replaceWith($svg);

        }, 'xml');
    });

function svg_mime_types( $mimes ) {
    $mimes['svg'] = 'image/svg+xml';
    return $mimes;
}
add_filter( 'upload_mimes', 'svg_mime_types' );

Additional fix:

function fix_mime_type_svg( $data = null, $file = null, $filename = null, $mimes = null ) {
    $ext = isset( $data['ext'] ) ? $data['ext'] : '';
    if ( strlen( $ext ) < 1 ) {
        $exploded = explode( '.', $filename );
        $ext      = strtolower( end( $exploded ) );
    }
    if ( $ext === 'svg' ) {
        $data['type'] = 'image/svg+xml';
        $data['ext']  = 'svg';
    } elseif ( $ext === 'svgz' ) {
        $data['type'] = 'image/svg+xml';
        $data['ext']  = 'svgz';
    }

    return $data;
}

add_filter( 'wp_check_filetype_and_ext', 'fix_mime_type_svg', 75, 4 );