Does it work with Cloudflare, or my host's own cache?

Yes. When one of these is active, every page NimbleCache clears is passed on to it: Cloudflare (through Cloudflare's own plugin, which clears a page only with APO on, or with a Cache Everything page rule; it does not read Cache Rules), Nginx Helper, Proxy Cache Purge, SiteGround Speed Optimizer, Breeze's Varnish, LiteSpeed Cache, Kinsta, and WP Engine for posts and pages. The message after Clear this page says which were asked. Most of them do not report back whether the clear worked, so "asked" is often what can be said; when the request to one fails or is refused, or it reports an error, the message says it could not be reached or answered with an error, and when one clears nothing, it says why.

The Integrations screen lists what it found, and says so when one of them clears nothing, with what to do. When the last Diagnostics check saw a cache NimbleCache cannot clear, it says what to do: for Cloudflare without its plugin, install Cloudflare's plugin or set Cloudflare to bypass the cache for pages; for a host's cache, clear it in the host's control panel after a change; for CloudFront, create an invalidation in your AWS account; for QUIC.cloud, LiteSpeed Cache clears it while active, or press Purge All CDN Cache on your QUIC.cloud dashboard. A cache whose header does not say whose it is, such as x-cache-status: HIT, is counted as cleared only when a plugin that clears a host's own cache is active, never because Cloudflare's plugin is. Also clear the cache in front on the same screen switches the clearing off.

When a post is saved and the cache in front could not be reached, whoever saved it is told on the next admin screen they open on that site, when they can edit posts: an editor or a shop manager as well as an administrator.

The other cache is cleared whole only when you press Clear the whole cache, in the toolbar or on the Dashboard, or run wp nimblecache purge. Behind Cloudflare that also clears the images, scripts and styles Cloudflare holds. When NimbleCache clears everything by itself, after a menu is saved for instance, it asks the other cache for the pages it held, up to 50, rather than emptying it and making every visitor fetch images and scripts from your server again.