Messages showing "Tap to Download" instead of images
Solution 1:
On the device with the issue, go to Settings -> Messages -> Send & Receive and then click on your Apple ID to sign out and then sign back in again. This seems to resolve the same issue for other people.
I think the reason for the issue you are experiencing is a corrupt file, by signing out and back in your device deletes a handful of files associated with your iMessage chats and regenerate them afresh from Apple's servers.