It sounds to me like you're running into a time issue. Specifically, WireGuard (and other VPNs) use time as part of the cryptographic process. If the time is wrong, the tunnel won't come up properly and the overall internet connection will appear to be down.
The reason this happens is due to the fact that the vast majority of consumer routers (especially all-in-one or embedded devices) do not have a persistent/real-time clock, so the initial date is derived from the most recent file that was touched on the filesystem, and then it is corrected once there is a successful sync with the network time servers via NTP. You must wait until this has completed before initiating Wireguard.