Welcome to the forums, BCH24. Nigel Owens, like all elite refs, has far too much on his plate managing the game itself to worry about whether the substitutions are being correctly managed. Throughout the English National League level, a 4th Official is assigned delegated responsibility for this. It's a demanding task - you do nothing for an hour except time and manage any yellow cards, and then all hell breaks loose with both sides wanting to make multiple substitutions over a very short period of time.
Above the national leagues, in the Championship and Premiership, more officials are added as more stuff needs to be managed accurately and timeously. In many cases, the referee has no need even to wear a watch.