It depends. I’ve done it a few different ways:

• YOLO: especially with thugs like PHP you only affect one page at a time and with low traffic the odds of a problem is small
• Maintenance page: temporarily show a page. Some servers like IIS have this built in. Otherwise it’s a simple update to httpd conf
• In a cluster environment, just take the node you’re updating out of rotation, and only update one nice at a time.
• Copy and switch like you suggested. Can be combined with any of the above and is a smart move if upload is slow or can be interrupted, or it’s cumbersome to restore the old files

I switched to self hosted Piwigo after Flickr started threatening to delete my photos a while back.

It had an extension that let me import all my photos from Flickr. Not sure if that still works after they changed hands.

It’s very easy to maintain; just click the update button in the Web UI. And it comes with a bunch of extensions.