The real keys to playing fast are in forming habits that serve to keep the game moving, and staying alert at the table. Once the auction is over, you can immediately place a card on the table. After your lead is turned face up, you can write the contract on your convention card as declarer studies dummy. Over the course of the night that might save a minute or two at most. The smoking ban in effect now at most clubs forces smokers to go outside or to a special room to smoke, where you can't hear the round being called.
Many clubs have washrooms, or coffee and other refreshments that are far away from the tables.
Some clubs even have less washroom space than they need and lineups result. None of these is any excuse for arriving at the table late. The only excuse for arriving at the table late is that you were still playing boards when the round was called. But some of us do so with such profusion that by the time the first bid is finally made, other tables have played three tricks already. The happy medium is to greet and pull cards from the slot simultaneously.