Automatic updates can take a few days to be available.
You should be able to go to the Apple App Store and search for Lightroom manually, it should let you download and install the update.
If you delete the app you will lose all local items. If you are subscribed customer who is fully synced that is not an issue as the items will resync from the cloud to your device. If you are subscribed but not fully synced, the unsynced material will be lost. If you are not subscribed - you will lose your images/presets.
If you are using the free version of Lightroom Mobile, please do not delete the app. The free app does not sync photos to cloud storage, and photos inside the free are app deleted when the app is deleted.
If you are a paid subscriber, and tapping the cloud button shows you are fully synced with no uploads pending, you should be able to uninstall and reinstall with no issue.
Before doing that, I think there is a simpler solution: Just search for the Lightroom app for iPad, and open the listing. You should see the "Open" button change to "Update." Our updates roll out in phases, and it can take days for the update to automatically occur, but searching for and opening our app listing should allow you to apply the update manually.
Thanks Charlie, I do have the paid version. When I looked up the app in the store, it showed as "open" and didn't change to "update". I didn't wait long, so if it takes more than a second or two I could've missed it. So I'll try that again later. If there's still no change, I'll try the uninstall procedure you suggested.