Perxel Image Optimizer

Perxel Image Optimizer converts your WordPress media library to WebP locally,
on your own server, and serves it – entirely from an admin page, built for shared
hosting.

• No third party. Conversion runs in PHP via WP_Image_Editor (GD/Imagick).
  Nothing is uploaded to a CDN or an external optimization service, and there is
  no API key, account or paid tier. Your images never leave your host.
• Free. The whole plugin, no upsell.
• Runs in the background. A light scan estimates the work, then a background
  job (Action Scheduler) converts the library newest-to-oldest, one calendar
  month at a time. Pause, resume, cancel, close the tab – progress is durable.
  Optional email report when a run finishes.
• Or run it Fast. An opt-in mode that drives the conversion from the open
  admin tab, so the server works continuously – much quicker on shared hosting
  with no cron. It throttles itself if the host pushes back, and stopping never
  loses work.
• Originals are untouched. foo.jpg gets a foo.jpg.webp sibling; the
  source file is never modified.
• Serving through a managed .htaccess block on Apache/LiteSpeed, with an
  to <picture> fallback on nginx or when .htaccess is not writable.
• Per-attachment Convert / Reconvert / Remove buttons in the Media library.
• New uploads are converted automatically, shortly after upload.

The admin area is two screens under Media > Optimization: an Optimization page (scan
→ prepare → live monitor) and a separate Settings page (environment,
conversion options, serving, notifications, cleanup).