Use sudo nixos-rebuild switch --flake .#<hostname> to rebuild the NixOS system with the specified hostname.
The #<hostname> part of that command can be skipped if the current hostname is already the same as one defined in the flake.
Updating the NixOS configuration will also update the home-manager configuration.
Likewise, use home-manager switch --flake . to switch the home-manager configuration only.
This could be useful on non-NixOS systems, though I haven't tested that yet.
Use nix flake update to update the package versions in flake.lock, then rebuild the system using the above command to update the system.