In most distros, it's systemctl start wg-quick@wg0 where wg0 is the name of the config file in /etc/wireguard
If so, then maybe double/triple check any firewalls / iptables rules. My VPS providers don't have any kind of firewall in front of the VM, but I'm not sure about Hetzner.
Maybe try stopping wireguard, starting a netcat listener on 51820 UDP and seeing if you can send to it from your homelab. This will validate that the UDP port is open and your lab can make the connection.