Hi kaylafromnewbury,
Thanks for coming to Apple Support Communities for help today with Live Photos that aren't playing inside Messages.
Check out: Take and edit Live Photos. Are you using the instructions "When you receive a Live Photo on your iPhone, iPad, or iPod touch via Messages, tap the photo to open it, then firmly press and hold to see it come to life."?
Are you able to Share to your Photos app and see them open there?
Have you tried watching on another network or cellular data?
Have you restarted your iPhone yet? Restart your iPhone, iPad, or iPod touch
Have you updated to iOS 13 yet? How to update to iOS 13.
Have a fantastic day!