If I reinstall Minecraft Pocket Edition, will my worlds be lost?

At least on iOS, deleting an app also deletes all its data, so deleting and reinstalling will delete your worlds unless your iOS device is backed up to the cloud (in which case your worlds will be stored on the cloud :D).

There is a way to save your worlds to a computer. On your PC, download and install a program called IExplorer. Connect your device to the PC and open IExplorer. You will see the files on your device, without a jailbreak.

Find the file in which the world is saved (Apps/com.mojang.minecraftpe/Documents/games/com.mojang/minecraftWorlds/), then Find the folder with the same name as your world that you want to save (notice they may not be in a specific order). Save this folder in a location of your choice on your PC, like the Desktop. You need to drag the folder to the desired location, for IExplorer does not have a right-click copy-paste feature.

Disconnect your device and reinstall Minecraft. Then reconnect your device and move the file back (drag) into the world saves folder (using IExplorer). You need to move the entire folder with nothing renamed.

Before you open any apps again make sure you disconnect your device. Then you will be able to get your world back.

This technique is useful for world backups and downloading and copying worlds between devices.

It is not necessary to install extra 3rd party applications to do something as simple as transferring some files. Apple has you covered with their own software. You can use the methods provided by first party sources, which tend to be more reliable, instead.

Just use the official method to restore your world. Mojang even supplies a guide on how to specifically copy minecraft data. The instructions boil down to:

First, transfer the files to your iCloud from your PC. You can use for example iCloud for Windows.

1. Open the 'Files' app.
2. Go the the 'Browse' screen. Go to 'iCloud Drive'
3. Tap on the folder containing your 'minecraftWorlds' folder. (For example 'Downloads')
4. Tap and hold on the minecraftWorlds folder.
5. Choose 'Copy' in the context menu.
6. Go back two levels up.
7. You should be in the main menu. Select 'My iPad'
8. Go to 'Minecraft/Games/com.Mojang' with three taps.
9. Tap and hold anywhere that isn't a folder or file in the main screen.
10. Select 'Paste' in this context menu.
11. A popup will ask you to confirm whether you want to replace. Choose 'Yes'. Important: You will override old savegame data on the iPad when you do.