I’d check Messages in iCloud first on both devices.
On the iPad, go to Settings > [your name] > iCloud > Messages and make sure Use on this iPad is turned on. Check the same setting on the iPhone. Both devices should also be signed in to the same Apple Account. Apple says Messages in iCloud keeps the full message history, including photos and other attachments, synced across your devices.
Also leave the iPad connected to Wi-Fi for a while in case it’s still downloading the attachments.
If those settings are already correct, does this happen with iMessages only, or also with SMS/MMS/RCS conversations? That would help narrow it down.