The following steps may help:
On your iPhone, in the Contacts app, check that there is a single contact card for each of your contacts and that each card contains that contact's full name and phone number, along with their email address.
If you have contacts from multiple sources and multiple entries for the same person have not been linked automatically (such that they are shown as a unified contact, with "Unified Info"), then manually link those contacts by following the instructions here:
Having made any changes in the Contacts app, on your iPhone, in the Watch app, go to: My Watch > General > Reset > tap on Reset Sync Data.
- Nothing will seem to happen, but the process will run in the background / invisibly to you, so allow some time afterwards for it to complete.
If the issue persists, it may then help to restart both your iPhone and your Apple Watch again, turning both off together, then restarting your iPhone first: