Make sure you have a working watch!! Or, if the show management supplies you with one, have a backup. Get the official "show time" from the judge or management. That will be the time that governs the day, even if it's fast or slow, compared to everyone else.
I like to have the riders lined out three deep - next in the ring, second to go, then third. Get the rider's numbers and be able to pick them out in the ring, so you can let them know the standings if they ask. When riding, I always liked to know the horse and rider that was in the ring before me so I could stop my warmup at the appropriate time and get ready. You may want to inform the riders of this - try it and see what the reception is like. If they don't want to be bothered, don't do it. Keep an eye on the show ring to let the riders know things are progressing. Don't stop and watch the tests, though! It's hard not to, but your job is in the warmup.
If someone is running late, just inform them of the time. Don't rush them - they're probably stressed already! Even if they just make it to the warmup arena as the rider before them is starting their test, they still have around five minutes to get ready. Or, if the rides are running ahead of schedule (and the riders will want to know what's going on when they hit the warmup), the rider does not have to go early. If the judge has asked and the rider is ready, they can go, but they don't have to ride early.
I always like to wish riders good luck and most seem to appreciate it. Don't get pressured into holding horses, coats, whips, etc., just because you happen to be standing there. This day, it's not your job.
Have fun - you can learn a lot in the warmup arena!