Caching images locally from external json feed

I'm currently working on a site that uses wordpress feeds to display posts. All seems to be working fine, however, the page load is very slow. I have implemented my own caching which means the the feeds will only be queried one every two hours (see code below).

$cacheFile =  'cached-data/data.txt';

$cacheData = file_get_contents($cacheFile);   
$two_hours = time() - 1;
$cacheTime = filemtime($cacheFile);

if ($cacheTime > $two_hours)
{
    $dataArray = $cacheData;
}
else
{
    //get data
    $historyData = file_get_contents('http://www.mysite.co.uk/feed/json?image=thumbnail&posts_per_page=5');
    $historyData = json_decode($historyData,true);

    $animalData = file_get_contents('http://www.mysiteothersite.co.uk/feed/json?image=thumbnail&posts_per_page=5');
    $animalData = json_decode($animalData,true);


    $dataArray = array_merge($historyData['posts'],$animalData['posts']);       

    $dataArray = json_encode($dataArray);

    $newCachFile = fopen($cacheFile, 'wb');
    fwrite($newCachFile, $dataArray);
    fclose($newCachFile);


}

$dataArray = json_decode($dataArray,ARRAY_A);

This speeds things up but the image load time is still rather slow. Therefore, I want to implement a function which downloads the images once an hour and modifies the the downloaded json so it replaces the image URLs to locally hosted images instead of the images being downloaded from their server.

Can anyone advice me on how to save the images locally?

Thanks