Je me sens vraiment perdu dans cette partie, est ce que c'est la bonne voie pour arriver à ce que je veux faire ?

<?php
session_start();
if(!empty($_SESSION['progress'])) {
    echo $_SESSION['progress'];
}else {
    echo 0;
}

le flush est inutile vu que ce n'est pas le but d'envoyer quelque chose à l'affichage (tu n'affiche rien d'ailleurs).

Pour la sauvegarde de la valeur précédente utilise plutôt une propriété de la classe que ta variable de méthode statique.

<?php
public function callback($download_size, $downloaded, $upload_size, $uploaded) {
        $_SESSION['progress'] = round($uploaded / $upload_size * 100, 1);
    }
?>

public function upload($file_path, $use_multiple_chunks = false, $chunk_temp_dir = '.', $size = 2097152, $replace_id = null)
    {
        if (!file_exists($file_path)) {
            return false;
        }

        // Figure out the filename and full size
        $path_parts = pathinfo($file_path);
        $file_name = $path_parts['basename'];
        $file_size = filesize($file_path);

        // Make sure we have enough room left in the user's quota
        $quota = $this->call('vimeo.videos.upload.getQuota');

        if ($quota->user->upload_space->free < $file_size) {
            throw new VimeoAPIException('The file is larger than the user\'s remaining quota.', 707);
        }

        // Get an upload ticket
        $params = array();

        if ($replace_id) {
            $params['video_id'] = $replace_id;
        }
        $rsp = $this->call('vimeo.videos.upload.getTicket', $params, 'GET', self::API_REST_URL, false);
        $ticket = $rsp->ticket->id;
        $endpoint = $rsp->ticket->endpoint;

        // Make sure we're allowed to upload this size file
        if ($file_size > $rsp->ticket->max_file_size) {
            throw new VimeoAPIException('File exceeds maximum allowed size.', 710);
        }

        // Split up the file if using multiple pieces
        $chunks = array();
        if ($use_multiple_chunks) {
            if (!is_writeable($chunk_temp_dir)) {
                throw new Exception('Could not write chunks. Make sure the specified folder has write access.');
            }

            // Create pieces
            $number_of_chunks = ceil(filesize($file_path) / $size);
            for ($i = 0; $i < $number_of_chunks; $i++) {
                $chunk_file_name = "{$chunk_temp_dir}/{$file_name}.{$i}";

                // Break it up
                $chunk = file_get_contents($file_path, FILE_BINARY, null, $i * $size, $size);
                $file = file_put_contents($chunk_file_name, $chunk);

                $chunks[] = array(
                    'file' => realpath($chunk_file_name),
                    'size' => filesize($chunk_file_name)
                );
            }
        }
        else {
            $chunks[] = array(
                'file' => realpath($file_path),
                'size' => filesize($file_path)
            );
        }

        // Upload each piece
        foreach ($chunks as $i => $chunk) {
            $params = array(
                'oauth_consumer_key'     => $this->_consumer_key,
                'oauth_token'            => $this->_token,
                'oauth_signature_method' => 'HMAC-SHA1',
                'oauth_timestamp'        => time(),
                'oauth_nonce'            => $this->_generateNonce(),
                'oauth_version'          => '1.0',
                'ticket_id'              => $ticket,
                'chunk_id'               => $i
            );

            // Generate the OAuth signature
            $params = array_merge($params, array(
                'oauth_signature' => $this->_generateSignature($params, 'POST', self::API_REST_URL),
                'file_data'       => '@'.$chunk['file'] // don't include the file in the signature
            ));

            // Post the file
            $curl = curl_init($endpoint);

            curl_setopt($curl,CURLOPT_NOPROGRESS,false);
            curl_setopt($curl,CURLOPT_PROGRESSFUNCTION, array($this, 'callback'));


            curl_setopt($curl, CURLOPT_RETURNTRANSFER, 1);
            curl_setopt($curl, CURLOPT_POST, 1);
            curl_setopt($curl, CURLOPT_POSTFIELDS, $params);
            $rsp = curl_exec($curl);
            curl_close($curl);
        }

        // Verify
        $verify = $this->call('vimeo.videos.upload.verifyChunks', array('ticket_id' => $ticket));

        // Make sure our file sizes match up
        foreach ($verify->ticket->chunks as $chunk_check) {
            $chunk = $chunks[$chunk_check->id];

            if ($chunk['size'] != $chunk_check->size) {
                // size incorrect, uh oh
                echo "Chunk {$chunk_check->id} is actually {$chunk['size']} but uploaded as {$chunk_check->size}<br>";
            }
        }

        // Complete the upload
        $complete = $this->call('vimeo.videos.upload.complete', array(
            'filename' => $file_name,
            'ticket_id' => $ticket
        ));

        // Clean up
        if (count($chunks) > 1) {
            foreach ($chunks as $chunk) {
                unlink($chunk['file']);
            }
        }

        // Confirmation successful, return video id
        if ($complete->stat == 'ok') {
            return $complete->ticket->video_id;
        }
        else if ($complete->err) {
            throw new VimeoAPIException($complete->err->msg, $complete->err->code);
        }
    }

public function callback($download_size, $downloaded, $upload_size, $uploaded) {
        static $last;
        if ($ind = @round($downloaded/$download_size*100, 1)) {
            if($last < $ind) {
                $_SESSION['progress'] = $ind;
                flush();
                $last = $ind;
            }
        } 
    }

public function ask()
    {
        $this->autoRender = false ;

        $prog = $_SESSION['progress'];

        return $prog;
    }