To avoid the queues, try using your Westminster Abbey tickets early, during off-peak hours. If you're visiting in July or August, arrive at the opening time, or about 20 minutes before.
Lunchtime is usually the busiest time of the day (just after the Changing of the Guard at Buckingham Palace).
Westminster Abbey closes later than usual on Wednesdays, at 18:00. Only individual visitors are allowed in at this time. Try to arrive by 16:00 and you should have plenty of time to explore in peace.
Worship takes place every day in the Abbey. There are usually four services daily at 07:30, 08:00, 12:30, and 17:00. On Sundays, there are six services at 08:00, 10:00, 11:15, 15:00, 17:45, and 18:30.